# How do you find the derivative of 1/(x+1)^2 ?

May 4, 2016

$\frac{d}{\mathrm{dx}} {\left(x + 1\right)}^{-} 2 = - \frac{2}{x + 1} ^ 3$

#### Explanation:

Method 1: Brute Force

Since this is a simple reciprocal, we can write it as a power and then use the chain rule:

$\frac{d}{\mathrm{dx}} {\left(x + 1\right)}^{-} 2 = - 2 {\left(x + 1\right)}^{-} 3 \frac{d}{\mathrm{dx}} \left(x + 1\right) = - \frac{2}{x + 1} ^ 3$

This answer seems so simple, we could have almost done it by inspection.

Method 2: Substitution

One way to prove this to ourselves is to do a substitution by setting

$y = x + 1 \implies \mathrm{dy} = \mathrm{dx} \implies \frac{\mathrm{dy}}{\mathrm{dx}} = 1$

which allows us to do the following substitution

$\frac{d}{\mathrm{dx}} {\left(x + 1\right)}^{-} 2 = \frac{d}{\mathrm{dy}} {y}^{-} 2 \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{d}{\mathrm{dy}} {y}^{-} 2$

now we end up with the very simple derivative in terms of $y$

$\frac{d}{\mathrm{dy}} {y}^{-} 2 = - 2 {y}^{-} 3$

which we can substitute back in for $x$ as

$\frac{d}{\mathrm{dx}} {\left(x + 1\right)}^{-} 2 = - \frac{2}{x + 1} ^ 3$

Just as we had before, but we can see why the problem was really very simple after all

Method 3: Inspection

The offset of $x$ by a constant doesn't affect the shape of the curve, i.e. the derivative. This is the same as thinking of a line translated in the horizontal direction, and the slope not being affected. Therefore, we can drop the constant when taking the derivative (translate horizontally) and then substitute it back when we're done (translate back horizontally)!