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) = nxn-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.
FAQs on Derivative of 1/x2
Answer: The derivative of 1/x2 is -2/x3.
Answer: The derivative of x2 is 2x.