Derivative of 1/x^2: Formula, Proof by First Principle

The derivative of 1/x² is equal to -2/x³. Note that 1/x² is an algebraic function. In this article, we will learn how to find the derivative of 1 divided by x² using the power rule, product rule, and the definition of derivatives.

Derivative of 1/x² Formula

The derivative of 1/x² can be expressed as d/dx(1/x²) or (1/x²)$’$. The derivative formula of 1 divided by x square is given below:

d/dx(1/x²) = -2/x³ or (1/x²)$’$ = -2/x³.

What is the Derivative of 1/x²?

Derivative of 1/x² by power rule: Let us first find the derivative of 1 by x² by the power rule of derivatives. Recall the power rule of derivatives: d/dx(xⁿ) = nx^n-1.

Follow the below steps to find the differentiation of 1 divided by x².

Express 1/x2 as a power of x	1/x2 = x-2
Differentiate both sides w.r.t. x	d/dx(1/x2) = d/dx(x-2)
Apply the power rule of derivatives	d/dx(1/x2) = d/dx(x-2) = -2x-2-1
Simplify	∴ d/dx(1/x2) = -2x-3 = -2/x3
Conclusion:	The derivative of 1/x2 by power rule is -2/x3.

Derivative of 1/x² 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:

$\dfrac{d}{dx}(f(x))$ $=\lim\limits_{h \to 0}\dfrac{f(x+h)-f(x)}{h}$

Put f(x) = 1/x²

So the derivative of 1/x² from first principle is

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

= $\lim\limits_{h \to 0} \dfrac{x^2-(x+h)^2}{hx^2(x+h)^2}$

= $\lim\limits_{h \to 0} \dfrac{x^2-(x^2+2xh+h^2)}{hx^2(x+h)^2}$

= $\lim\limits_{h \to 0} \dfrac{-h(2x+h)}{hx^2(x+h)^2}$

= $\lim\limits_{h \to 0} \dfrac{-(2x+h)}{x^2(x+h)^2}$

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

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

Thus, the derivative of 1/x² is equal to -2/x³ and this is obtained from the first principle of derivatives.

Also Read:

Derivative of 1/x:	-1/x2
Derivative of tan x:	sec2x
Derivative of esin x :	cos x esin x
Derivative of log 2x:	1/x

Derivative of 1/x² by Product Rule

Now, we will find the derivative of 1/x² by the substitution method together with the product rule of derivatives. For this let us put

z=1/x². We need to find dz/dx.

This implies that

zx²=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)

⇒ $z\cdot 2x+x^2\dfrac{dz}{dx}=0$

⇒ $x^2\dfrac{dz}{dx}=-2zx$

⇒ $\dfrac{dz}{dx}=-\dfrac{2z}{x}$

⇒ $\dfrac{dz}{dx}=-\dfrac{2}{x^3}$ as z=1/x²

So we have obtained the differentiation of 1/x² by the product rule which is -2/x³.

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