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 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.

$\dfrac{d}{dx}(x^n)=nx^{n-1}$

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 $(\sin x)^3$

Solution:

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)

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