Question

The derivative of $F\left[f\left\{\phi \left(x\right)\right\}\right]$ is

• A. $F\text{'}\left[f\left\{\phi \left(x\right)\right\}\right]$
• B. $F\text{'}\left[f\left\{\phi \left(x\right)\right\}\right]f\text{'}\left\{\phi \left(x\right)\right\}$
• C. $F\text{'}\left[f\left\{\phi \left(x\right)\right\}\right]f\text{'}\left\{\phi \left(x\right)\right\}\phi \text{'}\left(x\right)$
• D. none of these

Answer 1

The correct option is C

$F\text{'}\left[f\left\{\phi \left(x\right)\right\}\right]f\text{'}\left\{\phi \left(x\right)\right\}\phi \text{'}\left(x\right)$

Explanation for the correct option:

Simplifying $\frac{dF}{dx}=\frac{dF}{df}\chi \frac{df}{d\phi }\chi \frac{d\phi }{dx}$:

$\frac{dF}{dx}=\frac{dF}{df}\chi \frac{df}{d\phi }\chi \frac{d\phi }{dx}$

$$\begin{gathered} =\frac{d}{df}\left(F\left[f\left\{\phi \left(x\right)\right\}\right]\right)x\frac{d}{d\phi }\left(f\left\{\phi \left(x\right)\right\}\right)x\frac{d}{dx}\left(\phi \left(x\right)\right) \\ =F\text{'}\left[f\left\{\phi \left(x\right)\right\}\right]xf\text{'}\left\{\phi \left(x\right)\right\}x\phi \text{'}\left(x\right)\left[\frac{d}{dx}f\left(x\right)=f\text{'}\left(x\right)\right] \\ =F\text{'}\left[f\left\{\phi \left(x\right)\right\}\right]f\text{'}\left\{\phi \left(x\right)\right\}\phi \text{'}\left(x\right) \end{gathered}$$

Therefore, option (C) is correct.