Question

The derivatives of the function $f\left(x\right)=\cos \left(3x^2\right)$that follow are incorrect. What misconception occurs in each case?

$$\begin{gathered} \text{(a) Incorrect:}{f}^{\text{'}}\left(x\right)=(-\sin x)\left(3x^2\right)+\left(\cos x\right)\left(6x\right)\text{.} \\ \text{(b) Inсоrтесt:}{f}^{\text{'}}\left(x\right)=-\sin \left(6x\right)\text{.} \end{gathered}$$

Short answer

The answer is

Part (a) $\cos \left(3x^2\right)$where $\left(3x^2\right)$is a parameter, so product rule will not be applied.

Part (b) The chain rule will be applied outside the parameter and not inside.

Step-by-step solution

Part (a) Step 1. Given information.

The given function is$f\left(x\right)=\cos \left(3x^2\right).$

Part (a) Step 2. Explanation.

The function $f\left(x\right)=\cos \left(3x^2\right)$has a parameter as $\left(3x^2\right)$. So, product rule will not be applied.

In the given derivative,$(-\sin x)\left(3x^2\right)+\left(\cos x\right)\left(6x\right)$, product rule is applied.

Part (b) Step 1. Explanation.

In the given derivative, $-\sin \left(6x\right)$, chain rule is applied incorrectly.

It should have been applied outside the sin function.