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The question is to Differentiate the following with respect to x

\begin{align} Q1. \sin^3x \end{align}

The answer given is \begin{align} 3\sin^2x \cos x \end{align}

I am not sure how they got this answer so any help would be appreciated.

2 Answers 2

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Use the Chain Rule, where the derivative of $f(g(x))=f'(g(x))g'(x)$.

Let $f(x)=x^3$ and $g(x)=\sin(x)$, so that $f(g(x))=(\sin(x))^3$.

Apply the chain rule to get $3\sin^2(x)\cos(x)$.

  • 0
    So do I first apply the chain rule than use the product rule? – 2017-02-23
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    The product rule doesn't need to be used. You are not multiplying two functions together, but you are taking sin(x) and putting it into x^3 to get sin(x)^3 – 2017-02-23
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    The derivative of x^3 is 3x^2, so you have 3sin(x)^2 – 2017-02-23
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    Then, derivative of sin(x) is cos(x), so that is g'(x) that you tack onto the end – 2017-02-23
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HINT: Rewrite $\sin^3x$ as $(\sin x)^3$ and apply the chain rule.