3x²cos( x³ ) and 3sin²( x ) cos( x ) are the derivatives of the composite functions f(x) = sin(x³) and f(x) = sin³(x) respectively.
<h3>What are the derivative of f(x) = sin(x³) and f(x) = sin³(x)?</h3>
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;
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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