`y = root(3)(1 + 4x)` 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/

Textbook Question

Chapter 3, 3.4 - Problem 1 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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hkj1385's profile pic

hkj1385 | (Level 1) Assistant Educator

Posted on

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

Thus,

If y = (1+4x)^(1/3) ; then 

Let f(x) = x^(1/3)...........(1)

And g(x) = (1+4x).........(2)

Thus, f(g(x)) = (1+4x)^(1/3)  .................Answer

Now, dy/dx = y' = (1/3)*{(1 + 4x)^(-2/3)}*4

or, dy/dx = (4/3)*[(1+4x)^(-2/3)]

tiffanynelson's profile pic

tiffanynelson | In Training Educator

Posted on

Let it be noted that y is equivalent to `(1+4x)^(1/3)`

We see that the outer function y=f(u)= `u^(1/3)`

We see that the inner function u+g(x)=1+4x

The derivative of the inner function g'(x)=4

The derivative of the outer function f'(u)= ``

This is equivalent to `(1/3)u^(-2/3)`

To find the derivative of y, we simply multiply the two derivatives that we just found, f'(u) and g'(x) together and we replace the u with x. 

This gives 4`(1/3)x^(-2/3)`

This can also be written as  `(4)/(3x^(2/3))` 

balajia's profile pic

balajia | College Teacher | (Level 1) eNoter

Posted on

Let y=f(g(x))

y'=f'(g(x)).g'(x)

In the given question `y=(1+4x)^(1/3).`

Here `f(x)=(x)^(1/3)` and `g(x)=1+4x`

`y'=(1/3)(1+4x)^(1/3-1).(4)`

`=4/(3(1+4x)^(2/3))`

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