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Chain Rule of Differentiation

Differentiate...

Question

Differentiate $f (x)$ with respect to $g (x)$ for the following.
$f (x) = l o g_{e} x, g (x) = e^{x}$ .

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Solution

Given,
$f (x) = {log}_{e} x$ and $g (x) = e^{x}$ .
We are to find $\frac{d f (x)}{d g (x)}$ .
Now,
$\frac{d f (x)}{d x} = \frac{1}{x}$ and $\frac{d g (x)}{d x} = e^{x}$ .
Now,
$\frac{d f (x)}{d g (x)} = \frac{\frac{d f (x)}{d x}}{\frac{d g (x)}{d x}} = \frac{1}{x e^{x}}$ .

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