The derivative of 1/x3 is equal to -3/x4. Note that 1/x3 is an algebraic function. In this article, we will learn how to find the derivative of 1 divided by x3 using the power rule, product rule, and the definition of derivatives.
Derivative of 1/x3 Formula
The derivative of 1/x3 can be expressed mathematically as d/dx(1/x3) or (1/x3)$’$. The derivative formula of 1 divided by x cube is given below:
d/dx(1/x3) = -3/x4 or (1/x3)$’$ = -3/x4.
What is the Derivative of 1/x3?
Derivative of 1/x2 by power rule: At first, we find the derivative of 1 by x3 using the power rule of derivatives. Recall the power rule of derivatives: d/dx(xn) = nxn-1.
To find the differentiation of 1 divided by x3, follow the below steps as described in the table:
| Express 1/x3 as a power of x | 1/x3 = x-3 |
| Differentiate both sides w.r.t. x | d/dx(1/x3) = d/dx(x-3) |
| Apply the power rule of derivatives | d/dx(1/x3) = d/dx(x-3) = -3x-3-1 |
| Simplify | ∴ d/dx(1/x3) = -3x-4 = -3/x4 |
| Conclusion: | The derivative of 1/x3 by power rule is -3/x4. |
Derivative of 1/x3 by First Principle
The derivative of a function 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/x3
So the derivative of 1/x3 from first principle is
$\dfrac{d}{dx}\big(\dfrac{1}{x^3} \big)$ $=\lim\limits_{h \to 0}\dfrac{\frac{1}{(x+h)^3}-\frac{1}{x^3}}{h}$
= $\lim\limits_{h \to 0} \dfrac{x^3-(x+h)^3}{hx^3(x+h)^3}$
= $\lim\limits_{h \to 0} \dfrac{x^3-(x^3+3x^2h+3xh^2+h^3)}{hx^3(x+h)^3}$
= $\lim\limits_{h \to 0} \dfrac{-h(3x^2+3xh+h^2)}{hx^3(x+h)^3}$
= $\lim\limits_{h \to 0} \dfrac{-(3x^2+3xh+h^2)}{x^3(x+h)^3}$
= $\dfrac{-(3x^2+0+0)}{x^3(x+0)^3}$
= $\dfrac{-3x^2}{x^6}$ = $-\dfrac{3}{x^4}$.
Thus, the derivative of 1/x3 is equal to -3/x4 and this is obtained from the first principle of derivatives.
Also Read:
| Derivative of 1/x: | -1/x2 |
| Derivative of 1/x2: | -2/x3 |
| Derivative of esin x : | cos x esin x |
| Derivative of log 2x: | 1/x |
Derivative of 1/x3 by Product Rule
Next, using the substitution method together with the product rule of derivatives, we will find the derivative of 1/x3. For this let us put
z=1/x3. We need to evaluate dz/dx.
This implies that
zx3=1
Differentiating with respect to x, we get that
$\dfrac{d}{dx}(zx^3)=\dfrac{d}{dx}(1)$
⇒ $z\dfrac{d}{dx}(x^3)+x^3\dfrac{d}{dx}(z)=0$ (by the product rule of derivatives)
⇒ z ⋅ 3x2 + x3 $\dfrac{dz}{dx}$ =0
⇒ x3 $\dfrac{dz}{dx}$ = -3zx2
⇒ $\dfrac{dz}{dx}=-\dfrac{3z}{x}$
⇒ $\dfrac{dz}{dx}=-\dfrac{3}{x^4}$ as z=1/x3
So we have obtained the differentiation of 1/x3 by product rule which is -3/x4.
FAQs on Derivative of 1/x3
Answer: The derivative of 1/x3 is -3/x4.
Answer: The derivative of x3 is 3x2.
