Question

Derivative of log(1/x)

Answer 1

The derivative of log(1/x) is -1/x.

Given:

A function log(1/x).

To Find:

The derivative of log(1/x).

Solution:

To solve this problem, we will be applying the following concepts:

1) The derivative of a given function is essentially its rate of change concerning a variable. Geometrically, it is the same as calculating the slope of the tangent at a point to the given function.

2) Chain rule of derivatives:

The derivative of the composite of two functions f(g(x)) is calculated as:

$\frac{d}{dx}$ f(g(x)) = f(g(x))⋅g'(x).

3) Quotient Rule: The derivative of f(x)/g(x), where f and g are two functions is calculated as:

$\frac{d}{dx}$ $\frac{f(x)}{g(x)}$ = $\frac{f'(x)g(x)-g'(x)f(x)}{(g(x))^{2} }$

4) $\frac{d}{dx}$ log(x) = 1/x.

We have been given a function log(1/x). Note that this is a composite of two functions log(x) and (1/x).

Let us assume that f(x) = log (x) and g(x) = (1/x).

So, the derivative is calculated as:

$\frac{d}{dx}$(log(1/x)) = log'(1/x). (1/x)'.

⇒ $\frac{d}{dx}$(log(1/x)) = $\frac{1}{1/x}$ . (-$\frac{1}{x^{2} }$) = -1/x.

Hence, the derivative of log(1/x) is -1/x.

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