The derivative of 1/x^{3} is equal to -3/x^{4}. Note that 1/x^{3} is an algebraic function. In this article, we will learn how to find the derivative of 1 divided by x^{3} using the power rule, product rule, and the definition of derivatives.

Table of Contents

## Derivative of 1/x^{3} Formula

The derivative of 1/x^{3} can be expressed mathematically as d/dx(1/x^{3}) or (1/x^{3})$’$. The derivative formula of 1 divided by x cube is given below:

d/dx(1/x^{3}) = -3/x^{4} **or** (1/x^{3})$’$ = -3/x^{4}.

## What is the Derivative of 1/x^{3}?

Derivative of 1/x^{2} by power rule: At first, we find the derivative of 1 by x^{3} using the power rule of derivatives. Recall the power rule of derivatives: d/dx(x^{n}) = nx^{n-1}.

To find the differentiation of 1 divided by x^{3}, follow the below steps as described in the table:

Express 1/x^{3} as a power of x | 1/x^{3} = x^{-3} |

Differentiate both sides w.r.t. x | d/dx(1/x^{3}) = d/dx(x^{-3}) |

Apply the power rule of derivatives | d/dx(1/x^{3}) = d/dx(x^{-3}) = -3x^{-3-1} |

Simplify | ∴ d/dx(1/x^{3}) = -3x^{-4} = -3/x^{4} |

**Conclusion:** The derivative of 1/x^{3} by power rule is -3/x^{4}.

## Derivative of 1/x^{3} 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_{h→0} $\dfrac{f(x+h)-f(x)}{h}$

Put f(x) = 1/x^{3}

So the derivative of 1/x^{3} 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_{h→0} $\dfrac{x^3-(x+h)^3}{hx^3(x+h)^3}$

= lim_{h→0} $\dfrac{x^3-(x^3+3x^2h+3xh^2+h^3)}{hx^3(x+h)^3}$

= lim_{h→0} $\dfrac{-h(3x^2+3xh+h^2)}{hx^3(x+h)^3}$

= lim_{h→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/x^{3} is equal to -3/x^{4} and this is obtained from the first principle of derivatives.

**Also Read:**

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

Derivative of 1/x:^{2} | -2/x^{3} |

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

Derivative of log 2x: | 1/x |

## Derivative of 1/x^{3} by Product Rule

Next, using the substitution method together with the product rule of derivatives, we will find the derivative of 1/x^{3}. For this let us put

z=1/x^{3}. We need to evaluate dz/dx.

This implies that

zx^{3}=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 ⋅ 3x^{2} + x^{3 }$\dfrac{dz}{dx}$ =0

⇒ x^{3} $\dfrac{dz}{dx}$ = -3zx^{2}

⇒ $\dfrac{dz}{dx}=-\dfrac{3z}{x}$

⇒ $\dfrac{dz}{dx}=-\dfrac{3}{x^4}$ as z=1/x^{3}

So we have obtained the differentiation of 1/x^{3} by product rule which is -3/x^{4}.

## FAQs on Derivative of 1/x^{3}

**Q1: Find the derivative of 1/x**

^{3}.Answer: The derivative of 1/x^{3} is -3/x^{4}.

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

^{3}?Answer: The derivative of x^{3} is 3x^{2}.

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.