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Derivatives of Trigonometric Functions
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Step 1
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Solution:Step1:Recalltheproductruleforderivatives.Step4:Applytheproductrule.Usingtheproductrule,weget:f(x)=g(x)h(x)+g(x)h(x) =(15x2)(cos(x))+(5x3)(sin(x))Step5:Simplifytheexpression.f(x)=x2(15cos(x)5xsin(x))**Solution:** *Step 1: Recall the product rule for derivatives.* *Step 4: Apply the product rule.* Using the product rule, we get: \begin{align*} f'(x) &= g'(x)h(x) + g(x)h'(x) \ &= (15x^2)(cos(x)) + (5x^3)(-sin(x)) \end{align*} *Step 5: Simplify the expression.* \begin{align*} f'(x) &= x^2(15cos(x) - 5xsin(x)) \end{align*} **

Final Answer

** The derivative of the function
f(x)=5x3cos(x)f(x) = 5x^3 cos(x)
is
f(x)=x2(15cos(x)5xsin(x))f'(x) = x^2(15cos(x) - 5xsin(x))
.

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