Derivative of x^4 by First Principle, Power Rule

The derivative of x^4 is equal to 4x³ which can by proved by first principle and power rule. The formula of the derivative of x⁴ is given below.

$\dfrac{d}{dx}$(x⁴) = 4x³.

Let us now learn how to differentiate x to the power 4 by the power rule and the first principle of derivatives.

Derivative of x^4 by Power Rule

Recall the power rule of derivatives: the derivative of xⁿ by the power rule is

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

Putting n=4, we will get the derivative of x⁴ which is equal to

$\dfrac{d}{dx}(x^4)=4x^{4-1}=4x^3$.

Derivative of x^4 by First Principle

The derivative of f(x) by the first principle is as follows: $\frac{d}{dx}(f(x))$ $=\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$.

In this formula, we put f(x)=x⁴. So the differentiation of x⁴ using the first principle will be calculated as follows:

$\dfrac{d}{dx}(x^4)$ $=\lim\limits_{h \to 0} \dfrac{(x+h)^4-x^4}{h}$.

Expanding (x+h)⁴ using the formula (a+b)⁴=a⁴+b⁴+4a³b+6a²b²+4ab³, we get that

$\dfrac{d}{dx}(x^4)$ $=\lim\limits_{h \to 0}$ $\dfrac{x^4 + h^4 + 4x^3 h+6x^2h^2+4xh^3 -x^4}{h}$

= $\lim\limits_{h \to 0}$ $\dfrac{h^4 + 4x^3 h+6x^2h^2+4xh^3}{h}$

= $\lim\limits_{h \to 0}$ $\dfrac{h(h^3 + 4x^3 +6x^2h+4xh^2)}{h}$

= $\lim\limits_{h \to 0} h^3 + 4x^3 +6x^2h+4xh^2$

= $0^3 + 4x^3 +6x^2 \cdot 0+4x \cdot 0^2$

= 4x³.

So the derivative of x^4 by the first principle is 4x³.

Read Also: Derivative of 1/x by First Principle

Question-Answer

Question: Find the derivative of sin⁴x.

Answer:

Let z=sinx. Then by the chain rule of derivatives,

$\dfrac{d}{dx}(\sin^4 x)=\dfrac{d}{dz}(z^4) \cdot \dfrac{dz}{4x}$

= $4z^3 \cdot \cos x$

= 4sin³x cosx as z=sinx.

So the derivative of sin⁴x is equal to 4sin³x cosx.

ALSO READ:

Derivative of a^x

Derivative of 1/(1+x)

Derivative of e^3x

Derivative of $\sqrt{e^x}$

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