Question

I need help with part "C" and "D"

Answer 1

3x²cos( x³ ) and 3sin²( x ) cos( x ) are the derivatives of the composite functions f(x) = sin(x³) and f(x) = sin³(x) respectively.

What are the derivative of f(x) = sin(x³) and f(x) = sin³(x)?

Chain rule simply shows how to find the derivative of a composite function. It states that;

d/dx[f(g(x))] = f'(g(x))g'(x)

Given the data in the question;

• f(x) = sin(x³) = ?

• f(x) = sin³(x) = ?

First, we find the derivate of the composite function f(x) = sin(x³) using chain rule.

d/dx[f(g(x))] = f'(g(x))g'(x)

f(x) = sin(x)

g(x) = x³

Apply chain rule, set u as x³

d/du[ sin( u )] d/dx[ x³ ]

cos( u ) d/dx[ x³ ]

cos( x³ ) d/dx[ x³ ]

Now, differentiate using power rule.

d/dx[ xⁿ ] is nxⁿ⁻¹

cos( x³ ) d/dx[ x³ ]

In our case, n = 3

cos( x³ ) ( 3x² )

Reorder the factors

3x²cos( x³ )

Next, we find the derivative of f(x) = sin³(x)

d/dx[f(g(x))] = f'(g(x))g'(x)

f( x ) = x³

g( x ) = sin( x )

Apply chain rule, set u as sin( x )

d/du[ u³ ] d/dx[ sin( x )]

Now, differentiate using power rule.

d/dx[ xⁿ ] is nxⁿ⁻¹

d/du[ u³ ] d/dx[ sin( x )]

3u²  d/dx[ sin( x )]

Replace the u with sin( x )

3sin²(x)  d/dx[ sin( x )]
Derivative of sin x with respect to x is cos (x)

3sin²( x ) cos( x )

Therefore, the derivatives of the functions are 3x²cos( x³ ) and 3sin²( x ) cos( x ).

Learn more about chain rule here: brainly.com/question/2285262

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