Question

What is the derivative of $\cos ecx$?

Answer 1

Determine the derivative of $\cos ecx$.

$$\begin{gathered} \mathrm{Let},y=\mathrm{cosec}x \\ \Rightarrow y=\frac{1}{\sin x} \\ \Rightarrow y=(\sin x{)}^{-1} \end{gathered}$$

Use the formula of the chain rule

$$\begin{gathered} y=f\left(g\right(x\left)\right) \\ \Rightarrow \frac{dy}{dx}=f\text{'}\left(g\right(x\left)\right)\times g\text{'}\left(x\right) \end{gathered}$$

Now calculate the derivative:

$$\begin{gathered} \frac{dy}{dx}=-(\sin x{)}^{-2}\times \frac{d}{dx}(\sin x) \\ \Rightarrow \frac{dy}{dx}=-\frac{\cos x}{\sin x}\times \frac{1}{\sin x} \\ \Rightarrow \frac{dy}{dx}=-\cot x.\mathrm{cosec}x \end{gathered}$$

Hence, the derivative of $\cos ecx$ is$"-\cot x\mathrm{cosec}x"$.