Derivative of x^3 by First Principle, Power Rule

The derivative of x^3 is equal to 3x^2. The function x3 denotes the cube of x. In this post, we will find the derivative of x cube by different methods; for example, by the power rule, by the limit definition of derivatives.

Derivative of x^3

Derivative of x3 by Power Rule

To find the derivative of x cube, we will use the power rule of derivatives. The power rule is about the derivative of $x^n$ and the rule is given below.


Putting $n=3$ in the above rule, we will obtain the derivative of $x^2$.  Hence, it follows that

$\dfrac{d}{dx}(x^3)=3x^{3-1}$ $=3x^2$.

So the derivative of x3 by the power rule of derivatives is 3x2.

Next, we will find out the derivative of x3 from first principle, that is, using the limit definition of derivatives.

Derivative of x3 by First Principle

Recall the first principle of derivatives: The derivative of a function f(x) by first principle is given as follows.

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

Let $f(x)=x^3$

So the derivative of x cube by the first principle is equal to

$\dfrac{d}{dx}(x^3)$ $=\lim\limits_{h \to 0} \dfrac{(x+h)^3-x^3}{h}$

$=\lim\limits_{h \to 0} \dfrac{x^3+3x^2h+3xh^2+h^3-x^3}{h}$ using the algebraic identity $(a+b)^3$ $ = a^3+3a^2b+3ab^2+b^3$.

$=\lim\limits_{h \to 0} \dfrac{3x^2h+3xh^2+h^3}{h}$

$=\lim\limits_{h \to 0} \dfrac{h(3x^2+3xh+h^2)}{h}$

$=\lim\limits_{h \to 0} [3x^2+3xh+h^2]$

$=3x^2+3x \cdot 0+0^2$

$=3x^2+0 =3x^2$.

Hence, the derivative of x^3 by first principle is 3x^2.

Question 1: Find the derivative of $(\sin x)^3$


Let $z=\sin x$. So we have $\frac{dz}{dx}=\cos x$

Then by the chain rule of derivatives, the derivative of sinx cube is equal to

$\dfrac{d}{dx}(\sin^3 x)$ $=\dfrac{d}{dx}(z^3)$ $=\dfrac{d}{dz}(z^3) \cdot \dfrac{dz}{dx}$

$=3z^2 \cdot \cos x$

$=3\sin^2 x \cos x$ as z=sin x.

Thus, the derivative of (sin x)^3 is equal to 3sin2x cosx.

Also Read:

Derivative of 1/root(x)

Derivative of x

Derivative of x^2

Derivative of root x + 1 by root x

Derivative of 1/x^2 

Derivative of 1/(1+x^2)


Q1: What is the derivative of x3?

Answer: The derivative of x cube is equal to 3x2.

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