Question

If f (x) is an even function, then write whether f' (x) is even or odd.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have},f\left(x\right)\mathrm{is}\mathrm{an}\mathrm{even}\mathrm{function}. \\ \Rightarrow f\left(-x\right)=f\left(x\right) \end{gathered}$$

$$\begin{gathered} \Rightarrow \frac{d}{dx}\left\{f\left(-x\right)\right\}=\frac{d}{dx}\left\{f\left(x\right)\right\} \\ \Rightarrow f\text{'}\left(-x\right)\frac{d}{dx}\left(-x\right)=f\text{'}\left(x\right) \\ \Rightarrow f\text{'}\left(-x\right)\times \left(-1\right)=f\text{'}\left(x\right) \\ \Rightarrow -f\text{'}\left(-x\right)=f\text{'}\left(x\right) \\ \Rightarrow f\text{'}\left(-x\right)=-f\text{'}\left(x\right) \\ \mathrm{Thus},f\text{'}\left(x\right)\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{function}. \end{gathered}$$