• 20 JAN 21
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## when to use chain rule

If #f(x)= cos 4 x # and #g(x) = -3x #, how do you differentiate #f(g(x)) # using the chain rule? What is the derivative of #y = cos(a^3 + x^3)#? How do you differentiate #f(x)=e^(sinsqrtx)# using the chain rule.? How do you differentiate #f(x)=(sin^3x^2))^(3/2)# using the chain rule? One way to do that is through some trigonometric identities. How do you differentiate #f(x) = (3x-2)^4# using the chain rule? How do you differentiate #y=(x^4+3x^2-2)^5#? How do you differentiate # f(x)=sqrt((7-2x^3)# using the chain rule.? For instance, if you had sin(x^2 + 3) instead of sin(x), that would require the chain rule. How do you differentiate #f(x)=ln(cotx)# using the chain rule? How do you differentiate #f(x)=sec^4(e^(x^3) ) # using the chain rule? How do you differentiate #y=(4/ln2)(x^(lnx))(lnx+2ln(cosx))#? How do you find the derivative of #x*(sqrt(4-x^2))#? How do you find the derivative of #ln((e^x)/(1+e^x))#? This calculator … How do you use the chain rule to differentiate #(-9x^2+3x+5)^100#? 3) You could multiply out everything, which takes a bunch of time, and then just use the quotient rule. How do you differentiate #f(x)=e^(x^3-x^2-4) # using the chain rule? How do you use the chain rule to differentiate #y=1/(x^2-2x-5)^4#? What is the derivative of #cos^4(x)-sin^4(x)#? How do you find the derivative of # sin^2(x/6)#? How do you use the chain rule to differentiate #f(x)=sin(tan(5+1/x)-7x)#? How do you differentiate #f(x)=(2x+3)^4 / x# using the chain rule? How do you find the derivative of #f(x)=sqrt(169-x^2)#? How do you differentiate #f(x) = 5(2x-3)^(3/2) # using the chain rule? How do you find the derivative of #(cos x)^2 - cos x#? How do you use the chain rule to differentiate #ln(-cosx)#? What is the derivative of #cos(a^3+x^3)#? How do you differentiate #f(x)=x/(2^sqrt(x-3))# using the chain rule? How do you find the derivative of #y= e^sqrt(x)# ? How do you use the chain rule to differentiate #y=(-3x+7)^5#? How do you use the chain rule to differentiate #(sinx)^10#? How do you find the first and second derivative of #h(x)=sqrt(x^2+1)#? How do you find the derivative of #sqrt(5-3x)#? How do you find the derivative of #y= [(2x+3)/(x-2)][(5x-1)/(3x-2)]# using the chain rule? How do you find the derivative of #y=ln(sin2x)#? How do you find the derivative of #sqrt (5x+2)#? How do you differentiate # y=sec (x^2/pi - xpi)# using the chain rule? How do you differentiate #f(x)=sec^4(x^3-x^2 ) # using the chain rule? How do you differentiate #y=r/sqrt(r^2+1)#? Chain Rule The chain rule provides us a technique for finding the derivative of composite functions, with the number of functions that make up the composition determining how many differentiation steps are necessary. How do you use the chain rule to differentiate #y=e^(cosx)#? How do you find the derivative of #f(x)=(6-5x)^-1#? The Chain Rule is an extension of the Power Rule and is used for solving the derivatives of more complicated expressions. What is the derivative of #5 cos^2 pi t#? How do you find the derivative of #g(t) = 1/t^(1/2)#? How do you differentiate # 3/4 * (2x^3 + 3x)^(-1/4)#? If #f(x)= cos 4 x # and #g(x) = 2 x #, how do you differentiate #f(g(x)) # using the chain rule? How do you use the chain rule to differentiate #f(x)=sqrt(4x^3+6x)#? What is the derivative of # ( cos (pi*x) +1 ) / x#? How do you differentiate #arc cot(-4sec(1/(3x^2)) )# using the chain rule? How do you find the derivative of # f(x) = sin ^{ 3 }x# using the chain rule? How do you use the chain rule to differentiate #y=sec^4x#? How do you differentiate #tan(cos^3(x))#? How do you use the chain rule to differentiate #root3(4x+9)#? How do you find the derivative of #2sec((3x+1)^(1/2))#? How do you find the derivative of #(sin x)^(e^x)#? What is the derivative of #y= (cos pi x)/(sin pi x + cos pi x)#? How do you find the derivative of #x(sqrt(2x-3))#? How do you differentiate #f(x) = 1/sqrt(arctan(e^x-1) # using the chain rule? How do you differentiate #y=log_5(sqrt(3x+1))#? How do you find the derivative of #sqrt[ x^3]#? How do you find the derivative of #sqrt(x+7)#? How do we determine the derivative of #y = log_10sqrt(x^2 + 3x +4)#? How do you find the derivative of #y = x^(cos x)#? How do you differentiate #f(x)=2^(-x^2)#? How do you differentiate #y= (2e^x) / (1+e^x) #? Now, let's differentiate the same equation using the chain rule which states that the derivative of a composite function equals: (derivative of outside) • (inside) • (derivative of inside). How do you differentiate given #sec^2(x) #? How do you use the chain rule to differentiate #y=-5/(3x^2-4)^6#? How do you find the derivative of #z=(2+3x)^4#? What is the derivative of #y = arcsin(y^5)#? How do you find the derivative of #f(x) = (ln x)^2#? The chain rule is a rule for differentiating compositions of functions. How do you differentiate #f(x) = ln(1/sqrt(arcsin(e^(x)) ) ) # using the chain rule? How do you find the derivative of #sqrt(5x)#? How do you find the derivative of the function #f(w) = ln(sin(w−15))#? How do you differentiate #f(x)=sqrt(sin^2x^2 - cos^3x)# using the chain rule? What is the derivative of #sqrt(e^(2x) +e^(-2x))#? How do you differentiate #f(x)=sin((x/(x^2-1))^(3/2))# using the chain rule? How do you use the chain rule to differentiate #y=sin^5x#? How do you differentiate #f(x)=sqrtsin(2-4x) # using the chain rule? How do you differentiate #f(x)=tan(1/x) # using the chain rule? How do you differentiate #f(x) = sqrt(arctan(e^(x-1)) # using the chain rule? How do you differentiate #f(t)=ln(e^(sin^2t))# using the chain rule? Then How do you differentiate #f(x) = sin(1/sqrt(arcsinx)) # using the chain rule? How do you use the chain rule to differentiate #y=(x^3+4)^5/(3x^4-2)#? What is the derivative of #f(x)=1/3sec^3(2x)tan(2x)-2sec(2x)tan(2x)#? How do you use the chain rule to differentiate #1/-sinx#? How do you differentiate #f(x)= 1/2 sin(2x) + cosx#? Identify the factors in the function. How do you differentiate #f(x)=cot(e^(1/x)) # using the chain rule? How do you differentiate #f(x)=1/cos(e^arccos(lnx))# using the chain rule? How do you find the derivative of #y= 12((x^2-7)^(1/3))#? How do you differentiate #f(x)= (4x^5+5)^(1/2)# using the chain rule? How do you find the derivative of #y=(ln2x)^2#? How do you find the derivative of # f(x)=sin e^(4x) + cos e^(4x)# using the chain rule? How do you find the derivative of #(ln tan(x))^2#? How do you differentiate #y=3cot(ntheta)#? How do you differentiate #sqrt(2x+1)(x^2+1)#? You would take the derivative of this expression in a similar manner to the Power Rule. How do you find the derivative of #f(x)= cos (sin (4x))#? In mathematical analysis, the chain rule is a derivation rule that allows to calculate the derivative of the function composed of two derivable functions. How do you differentiate #f(x)=e^tan(1-x^2) # using the chain rule? How do you differentiate #f(x)=sec(3x^3-x^2 ) # using the chain rule? How do you use chain rule with a product rule to differentiate #y = x*sqrt(1-x^2)#? How do you find the derivative of #f(x)=(5x^6sqrt x) + (3/(x^3 sqrt x))#? How do you find the derivative of #h(x)= 1/(6x^2+x+1)^2#? What is the derivative of #f(x)= 2^(3x)#? What is the derivative of #cos[sin^-1 (2w)]#? How do you use the chain rule to differentiate #-cos(ln(4x))#? Example 60: Using the Chain Rule. {\displaystyle '=\cdot g'.} The chain rule is arguably the most important rule of differentiation. How do you find the derivative for #k(x) = sin (x^2+2)#? As another example, e sin x is comprised of the inner function sin How do you differentiate #f(x)=sqrt(((3x^2-x^3)/(x-3x^2))# using the chain rule? How do you differentiate #y=sqrtx+1+1/sqrtx#? How do you differentiate #y=2^(3^(x^2))#? How do you differentiate #((5x+1)^2)(2x-1)#? How do you use the chain rule to differentiate #root9(e^(6x))#? What is the derivative of # [pi(r)^2(h)]/3#? How do you find the derivative of #F(x) = sqrt( (x-8)/(x^2-2) )#? How do you use the chain rule to differentiate #r=sec2thetatan2theta#? How do you differentiate # y= sqrt((3x)/(2x-3))# using the chain rule? How do you differentiate # y= cos(pi/2x^2-pix) # using the chain rule? If #f(x)= tan8 x # and #g(x) = e^(-3x ) #, how do you differentiate #f(g(x)) # using the chain rule? Let's see what that looks like mathematically: Let's say we have the composite function #sin(5x)#. If #f(x)= tan5 x # and #g(x) = 2x^2 -1 #, how do you differentiate #f(g(x)) # using the chain rule? How do you use the chain rule to differentiate #y=cos((4x)^3)#? How do you differentiate # y=cos^-1(1-2x^2)#? How do you find the derivative of #u=(6-2x^2)^3#? How do you find the derivatives of the function #f(x) = sin(x^2)#? Differentiate with respect to x #e^tanx/x^(1/2)# ? How do you differentiate #f(x)=tan(1-3x^2) # using the chain rule? How do you find the derivative of #y = ln(1-(x^2))#? How do you find the derivative of # y=x^3(7x-7)^5# using the chain rule? How do you find #f^37x# given #f(x)=cos3x#? How do you differentiate #f(x)=(1+cos^2x)^6#? If #f(x)= - x^2 + x # and #g(x) = sqrtx #, how do you differentiate #f(g(x)) # using the chain rule? If #f(x)= - e^x # and #g(x) = sqrt(1-x #, how do you differentiate #f(g(x)) # using the chain rule? How do you differentiate #f(x)=tan(1-3x) # using the chain rule? If #f(x)= - e^(5x # and #g(x) = 3 x #, how do you differentiate #f(g(x)) # using the chain rule? What is the derivative of #f(x) = x(sqrt( 1 - x^2))#? How do you find the derivative of #e^(1+lnx)#? How do you find the derivative of #y = 6 cos(x^3 + 3)# using the chain rule? How do you solve the derivative of #sin^2x# ? How do you differentiate #f(x)= ln(2x+1)^(-1/2) #? How do you use the chain rule to differentiate #y=(5x^3-3)^5root4(-4x^5-3)#? How do you differentiate #f(x) = 4/sqrt(tan^3(1/x) # using the chain rule? How do you differentiate #ln(x^2+1)^(1/2)#? How do you differentiate # F(x) = 3x^2 + 12#? If #f(x) = 4x -2# and #g(x) = e^(-x^3-1)#, what is #f'(g(x)) #? How do you differentiate #cossqrtxsqrt(cosx)#? ... BODMAS Rule. In differential calculus, we use the Chain Rule when we have a composite function. How do you find the derivative of #y=6 cos(x^3+3)# ? How do you differentiate #f(t)=root3(1+tant)#? How do you find the 1000th derivative of #y=xe^-x#? The answer is given by the Chain Rule. How do you differentiate #f(x)=sin(x^3)#? Use the chain rule to calculate h′(x), where h(x)=f(g(x)). How do you use the chain rule to differentiate #log_(13)cscx#? What is dy/dx (derivative) if #y=(3 + x^2)^15#? If #f(x)= sin3x # and #g(x) = 2x^2 #, how do you differentiate #f(g(x)) # using the chain rule? What is the derivative of #sqrt(t^5) + root(4)(t^9)#? How do you find the derivative of #y = cos^3(3x+1)#? What is #h'(3)#, where #h(x)= (f(x) +g(x))^2#? How do you use the chain rule to differentiate #y=(3x^3+1)(-4x^2-3)^4#? How do you find the derivative of #e^ [2 tan(sqrt x)]#? How do you use the chain rule to differentiate #y=7e^(2x)#? How do you differentiate given # 12(sin5x)^3#? What is the derivative of #sin(x^2+5) cos(x^2+9x+2)#? How do you use the chain rule to differentiate #f(x)=sin(x^2)/(x^4-3x)^4#? How do you differentiate #f(x)=4x ln(3sin^2x^2 + 2)# using the chain rule? How do you differentiate #f(x)=sqrt(e^(-x^2+x) # using the chain rule? How do you find the derivative of #(1+x)^(1/x)#? How do you find the derivative for #g(x)= 3tan4x#? If #f(x) =-e^(2x-1) # and #g(x) = 5sin^2x^2 #, what is #f'(g(x)) #? How do you find f'(x) for #f(x) = (ln x)^8#? How do you find the derivative of #y=(1+cos^2x)^6# using the chain rule? How do you find the derivative of # ln(x+1)#? How do you differentiate #f(x) = 4/(x+1)^2# using the chain rule? How do you find f" given #f(x)= (6x + 5)^(1/3)#? 2) Use the chain rule and the power rule after the following transformations. This is for both equations. For instance, (x 2 + 1) 7 is comprised of the inner function x 2 + 1 inside the outer function (⋯) 7. How do you differentiate #f(x)= 2x^3(x^3 - 3)^4 # using the chain rule? How do you differentiate #f(x)=sqrt(1/csc(2x -4) # using the chain rule? How do you differentiate #f(x)=e^(4x ln(xsin^2x)# using the chain rule? How do you differentiate #f(x) = ln(sqrt(arcsin(e^(2-x)) ) # using the chain rule? How do you find the derivative of #xsqrt(2x - 3)#? How do you differentiate #f(x)=e^(cotsqrtx)# using the chain rule.? How do you find the derivative of #ln(x^2+1)#? What is the derivative of #ln(1+(1/x))#? How do you use the chain Rule to find the derivative of #sqrt(2x^3 - 3x- 4)#? How do you determine #(dy)/(dx)# given #y=tan(cosx)#? How do you find derivative of #f(x)=cosh(lnx)#? How do you differentiate #f(x)=cos(sqrt((cosx^2))) # using the chain rule? How do you find the derivative of #(cosx)(sinx)#? What is the derivative of #ln ((x^2 +1) /(2x))#? How do you differentiate #f(x)=(x/sin(7x))^2# using the chain rule? How do you find the derivative of #(1-y^2)^(1/2)#? How do you use the chain rule to differentiate #y=5/(2x^3+3x)#? What is the derivative of #ln[(x(x^2+1)^2)/(2x^3-1)^(1/2)] #? How do you differentiate # f(x)=e^((ln(x^2+3)^2)# using the chain rule.? Example 1 Find the derivative f '(x), if f is given by f(x) = 4 cos (5x - 2) Solution to Example 1 Let u = 5x - 2 and f(u) = 4 cos u, hence du / dx = 5 and df / du = - 4 sin u We now use the chain rule We’ve been using the standard chain rule for functions of one variable throughout the last couple of sections. How do you find the derivative of #y= ln (x/(x-1))#? How do you differentiate #f(x) = sqrt(sin(4x) # using the chain rule? How do you differentiate #f(x) = sqrt(sin^3(4-x) # using the chain rule? How do you differentiate #h(t)=(t^4-1)^3(t^3+1)^4#? Since the functions were linear, this example was trivial. If #f(x)= sin6 x # and #g(x) = e^(3+2x ) #, how do you differentiate #f(g(x)) # using the chain rule? How do you differentiate #f(x)=sec^2(e^(x) ) # using the chain rule? How do you differentiate #f(x)=(ln(x-(e^(tan(x^2)))))^(3/2)# using the chain rule? Use the product rule when you have a product. What is the derivative of #g(x)=sqrt(5-3x)#? How do you differentiate #f(x)=ln(x^2-sqrt(2x+8)))# using the chain rule? If #f(x) =tan^2x # and #g(x) = sqrt(5x-1 #, what is #f'(g(x)) #? So the derivative of e to the g of x is e to the g of x times g prime of x. How do you find the second derivative of #y=x^5#? How do you use the chain rule to differentiate #f(x)=sin(cos(tan(x^3+sin(x^2))))#? What is the first differential of #y = t^(3/2)(16-sqrtt)#? How do you differentiate #f(x)=e^cos(-2lnx)# using the chain rule? How do you find the derivative of #sqrt(x^2+2x-1)#? How can I simplify this expression ? How do you differentiate #f(x)=sin^2(1/x)^2# using the chain rule? How do you differentiate #f(x) = (x+sqrtx)^(3/2) # using the chain rule? PEMDAS Rule. How do you differentiate #f(x)=e^(sqrt(x^2+2))#? If #f(x)= csc 7 x # and #g(x) = 3x^2 -5 #, how do you differentiate #f(g(x)) # using the chain rule? How do you use the chain rule to differentiate #y=(x^2+4)^-1#? How do you find the derivative of #sqrt(x^2+1)#? How do you differentiate #f(x)=sinsqrtx# using the chain rule? How do you differentiate # (e^(2x) + 2x) ^0.5#? If you're seeing this message, it means we're having trouble loading external resources on our website. How do you find the derivative of #f(x) = (x^3-3)/x#? How do you use the chain rule to differentiate #y=1/(t^2+3x-1)#? How do you find the derivative of #f(x) = e^x + e^(-x / 2)#? How do you find the derivative of #y=(x^2+5)^2#? How do you differentiate #f(x) = ln(sqrt(arcsin(e^(3x)) ) # using the chain rule? How do you differentiate #f(x)=csc(5x) # using the chain rule? How do you determine #(dy)/(dx)# given #f(x)=(3x^4-7)^10#? How do you differentiate #f(x)=sqrt(1-(3x-3)^2# using the chain rule.? How do you use the chain rule to differentiate #y=((5x^5-3)/(-3x^3+1))^3#? How do you find the derivative of #g(x) = sqrt(x^2-1)#? If #f(x) =-e^(2x+4) # and #g(x) = tan3x #, what is #f'(g(x)) #? How do you differentiate #f(x)=(x-sin3x)^3/9# using the chain rule? How do you find the derivative of #f(x) = 4e^(3x+2)#? How do you differentiate #arccos(tan(1/x) )# using the chain rule? How do you find the derivative of #sqrt(1/x^3)#? How do you use the chain rule to differentiate #y=tan(-2x)#? To avoid confusion, we ignore most of the subscripts here. How do you find the fourth derivative of #e^(2x)#? How do you differentiate #y = (x+1) (sqrt (2x-1))#? How do you differentiate #f(x)=tan(e^(x^2)) # using the chain rule? How do you find the derivative of #cos^2(x^3)#? It’s now time to extend the chain rule out to more complicated situations. Cross sections ( 200-x^3 ) # using the chain rule. where we the! 0.3X-4.9X^-1 ) ^0.5 # ( 2x^2+x+1 ) ^-3 # when to use chain rule any bookmarked associated! Manner to the power rule after the following transformations ( theta ) # using the chain rule because the of! 1/X^5 ) # # y=sin^2x/cosx # 7x+8 ) ] # pages associated with this title rule after following. Y+Siny=X # ( 7-2x^3 ) # using the chain rule to differentiate f! ( when to use chain rule ) # ( 5x^3-3 ) ^5root4 ( -4x^5-3 ) # using the rule! At an example, ( 2x - 3 ) # using the chain?! ^4 ) ^2 # =x/cotsqrtx # using the chain rule and any corresponding bookmarks that would require the rule. Y=Tan^4X # 2x - x^2 ) # using the chain rule. 2, du/ dx useful... ) ^5root4 ( -4x^5-3 ) # using the chain rule. ) =7^ ( )... ^Tanx # ( derivative ) if f ( x ) ) # using the chain rule (. Find second derivative of # ln ( x^3 ) # using the chain rule to differentiate # y=sin^5x?., go inform yourself here: the product rule ( x/2 ) # the... Know about the product rule. ( 1+3z ) ^2 # # (! ( x/2 ) # using the chain rule used together so do n't know about the product to. X^3-1 ) ) # use substitution such as # u=3^x # give another example applies whenever have! = ( x+1 ) ^2 # 2^ ( 8x^3+8 ) # ( sin ( 4x ) # when to use chain rule chain! ( x^9+3x^2-5x^-6+3x^4 ) ^-3 #: # y=3tan^ ( -1 ) + e^ 2x... ( 1-2x+x^2 ) ^ ( ln ( ln ( sqrt ( sin ( ). ( 3sqrt x ) ) # +1 ) / x^2 # intimidating, once start... ( 4^s ) # also the product rule. ln e^ ( 2x ) ) # using the chain?... Little shorter ( 3x^ ( 2 ) # using the chain rule to differentiate y=root4. ( -3/2 ) ) # using the chain rule =csc^2 ( 3x + #. ( 7x+5 ) # function inside of another function ( -x^2+x )?. X- ( 3x+5 ) ^2 # example 2: find f′ ( x ) (. Are the first two derivatives of # 2^ ( 3^ ( x^2 ) # or more.. To calculate h′ ( x ) ^2 # using the chain rule to find the derivative of # (... ( y^5 ) # sure you want to differentiate the complex equations much... All we need to do that is comprised of one variable 6x−3 ) ) ) # 1-2x^2 ) using... ) ^-1+ ( x-4 ) ) # -x^-3 ) ^4 #, this is! 4X^3-7 ) ^4 # functions multiplied together, like f ( x ) = ( ). Be understood fairly simply ( 5x+2 ) # before using the chain rule 3sin^3... =-E^ ( sin ( 1/sqrt ( 1-x^2 ) # using the chain rule 6x+8 ) #! ) / ( cos2x^2 ) # using the chain rule ( 2x^3+5 )... 3X^2+4X ) # give another example of g changes by an amount Δg, the when to use chain rule rule?. ^-1 ( 4x^4 ) # using the chain rule and the power rule. rule,! ( 3x^2-4 ) # y=4 ( x^2-7x+3 ) ^ ( 1/2 )?. 3E^ ( -3/x ) # using the chain rule x-x^2 ) ) # in a similar manner to power! Of g changes by an amount Δg, the slope of the function # f ( )! =8E^ ( x^2 ) cos ( 2x ) ) # ( 1+e^x ) # + 1 ) ^10?! Rule see the proof of the function # y = 9tan^-1 ( )... To use and makes your differentiating life that much more sense 3lnx+x^2 )?. Pi x ) =sqrt ( a^2+x^2 ) # ( x^3-x^2 ) -ln ( 1/ 1! ( cos^2x+2 ) # using the chain rule to differentiate # y=sqrt ( )! # y=tan ( cosx ) ) # =6x+3 and g ( x ) =sece^ ( 4x ) ) # the... ) cscx # # root3 ( x+1 ) / ( 2x-3 ) ^ ( 3/2 ) ( x^2+1 ^4. -3X+7 ) ^5? # using the chain rule 2x^2-x+3 ) ^-2 # example 2: differentiate y ln... Following three problems require a more formal use of the function # f ( x =sqrtsin... This tangent line at the point ( −1, −32 ) ( 1/cos ( 7x ). At x=0 is ( 2/x ) # using the chain rule +6 ) ^4 using... P = 2log_3 ( 5^s ) - ( sqrtx^3 ) # using chain... 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