`y = (2x^3 + 5)^4` Write the composite function in the form f(g(x)). Identify the inner function u = g(x) and the outer function y =f(u). Then find the derivative dy/dx

Textbook Question

Chapter 3, 3.4 - Problem 2 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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hkj1385's profile pic

hkj1385 | (Level 1) Assistant Educator

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Notes:- 1) If y = x^n ; then dy/dx = n*x^(n-1)

2) If a function to be differentiated contains sub-functions, then the differentiation of the last function is done first and so on

Now, given 

y = {2*(x^3) + 5}^4

Let f(x) = x^4 ..........(outer function)

and g(x) = {2*(x^3) + 5}.........(inner function)

Thus, f(g(x)) = {2*(x^3) + 5}^4.............answer

Now, 

y = {2*(x^3) + 5}^4

thus, dy/dx = [4*{2*(x^3) + 5}^3]*[6*x^2]

or, dy/dx = {24*(x^2)}*[{2*(x^3) + 5}^3]

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balajia | College Teacher | (Level 1) eNoter

Posted on

Given function  `y=(2x^3+5)^4`

This is in the form `y=f(g(x))`

Here `f(x)=x^4` and `g(x)=2x^3+5`

`y'=4(2x^3+5)^3.(6x^2)`

`y'=24x^2(2x^3+5)^3`

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