The derivative of 1/x^{2} is equal to -2/x^{3}. In this article, we will learn how to find the derivative of 1 divided by x^{2} using the power rule, product rule, and the definition of derivatives.

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## Derivative of 1/x^{2} Formula

Note that 1/x^{2} is an algebraic function. The derivative of 1/x^{2} can be expressed as d/dx(1/x^{2}) or (1/x^{2})$’$. The derivative formula of 1 divided by x square is given below:

d/dx(1/x^{2}) = -2/x^{3} **or** (1/x^{2})$’$ = -2/x^{3}.

## What is the Derivative of 1/x^{2}?

Derivative of 1/x^{2} by power rule: Let us first find the derivative of 1 by x^{2} by the power rule of derivatives. Recall the power rule of derivatives: d/dx(x^{n}) = nx^{n-1}.

Follow the below steps to find the differentiation of 1 divided by x^{2}.

Express 1/x^{2} as a power of x | 1/x^{2} = x^{-2} |

Differentiate both sides w.r.t. x | d/dx(1/x^{2}) = d/dx(x^{-2}) |

Apply the power rule of derivatives | d/dx(1/x^{2}) = d/dx(x^{-2}) = -2x^{-2-1} |

Simplify | ∴ d/dx(1/x^{2}) = -2x^{-3} = -2/x^{3} |

**Conclusion:** The derivative of 1/x^{2} by power rule is -2/x^{3}.

## Derivative of 1/x^{2} by First Principle

If f(x) is a function of real variable x, then the derivative of f(x) by the first principle is given by the following limit formula:

$\dfrac{d}{dx}(f(x))$ = lim_{h→0} $\dfrac{f(x+h)-f(x)}{h}$

Put f(x) = 1/x^{2}

So the derivative of 1/x^{2} from first principle is

$\dfrac{d}{dx}\big(\dfrac{1}{x^2} \big)$ = lim_{h→0} $\dfrac{\frac{1}{(x+h)^2}-\frac{1}{x^2}}{h}$

= lim_{h→0} $\dfrac{x^2-(x+h)^2}{hx^2(x+h)^2}$

= lim_{h→0} $\dfrac{x^2-(x^2+2xh+h^2)}{hx^2(x+h)^2}$

= lim_{h→0} $\dfrac{-h(2x+h)}{hx^2(x+h)^2}$

= lim_{h→0} $\dfrac{-(2x+h)}{x^2(x+h)^2}$

= $\dfrac{-(2x+0)}{x^2(x+0)^2}$

= $\dfrac{-2x}{x^4}$ = $-\dfrac{2}{x^3}$.

Thus, the derivative of 1/x^{2} is equal to -2/x^{3} and this is obtained from the first principle of derivatives.

**Also Read:**

Derivative of 1/x: | -1/x^{2} |

Derivative of tan x: | sec^{2}x |

Derivative of e :^{sin x} | cos x e^{sin x} |

Derivative of log 2x: | 1/x |

## Derivative of 1/x^{2} by Product Rule

Now, we will find the derivative of 1/x^{2} by the substitution method together with the product rule of derivatives. For this let us put

z=1/x^{2}. We need to find dz/dx.

This implies that

zx^{2}=1

Differentiating with respect to x, we get that

$\dfrac{d}{dx}(zx^2)=\dfrac{d}{dx}(1)$

⇒ $z\dfrac{d}{dx}(x^2)+x^2\dfrac{d}{dx}(z)=0$ (by the product rule of derivatives)

⇒ z⋅2x + x^{2} $\dfrac{dz}{dx}$ = 0

⇒ $x^2\dfrac{dz}{dx}=-2zx$

⇒ $\dfrac{dz}{dx}=-\dfrac{2z}{x}$

⇒ $\dfrac{dz}{dx}=-\dfrac{2}{x^3}$ as z=1/x^{2}

So we have obtained the differentiation of 1/x^{2} by the product rule which is -2/x^{3}.

**Also Read:** How to Differentiate 1/x^{3}?

## Solved Problems

**Question:** Find the derivative of 1/sin^{2}x.

**Answer:**

Let z=sinx.

So dz/dx = cosx.

So by the chain rule, the derivative of 1/sin^{2}x is equal to

d/dx (1/sin^{2}x)

= $\dfrac{d}{dz}(\dfrac{1}{z^2}) \times \dfrac{dz}{dx}$

= $-\dfrac{2}{z^3} \times \cos x$, by the above differentiation rule of 1/x^{2}.

= $-\dfrac{2\cos x}{\sin^3 x}$ as z=sinx.

So the derivative of 1/sin^{2}x is equal to -2cosx/sin^{3}x, obtained by the chain rule of derivatives.

## FAQs on Derivative of 1/x^{2}

**Q1: What is the derivative of 1/x**

^{2}?Answer: The derivative of 1/x^{2} (1 over x square) is equal to -2/x^{3}.

**Q2: What is the derivative of x**

^{2}?Answer: The derivative of x^{2} (x square) is 2x.

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.