Derivative of a function with respect to another function. [duplicate]

functions derivatives self-learning partial-derivative

Solution 1:

$$\frac{dg(x)}{df(x)} = \frac{dg(x)}{dx} \cdot \frac{1}{f'(x)} = \frac{g'(x)}{f'(x)}$$

In your example,

$$g'(x) = 2f'(x) + 1 + \frac{f'(x)}{f(x)}$$

So:

$$\frac{dg(x)}{df(x)} = \frac{2f'(x) + 1 + \frac{f'(x)}{f(x)}}{f'(x)} = 2 + \frac{1}{f'(x)} + \frac{1}{f(x)}$$

Solution 2:

You can not. You have to derivate $f(x)$ as function.

$g'(x) = 2f'(x) + 1 + {f'(x) \over f(x)}$

EDIT: Sorry, That would make $dg(x) \over dx$, Deepak is right.

Solution 3:

You could if it were a function of $f(x)$ But it's not, due to the $x$ term.

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