Derivative of x^5 by First Principle, Power Rule

The derivative of x^5 (x to the power 5) is equal to 5x4. Here, we will learn how to find the derivative of x5 using the power rule and the first principle of derivatives.

The derivative of x5 is denoted by d/dx (x5). The derivative formula of x5 is as follows:

$\dfrac{d}{dx}(x^5)=5x^4$.

Derivative of x^5 by Power Rule

We know that the derivative of xn by the power rule is given by the formula:

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

Put n=5.

So the derivative of x5 by the power rule will be equal to $\dfrac{d}{dx}(x^5)$ = 5x5-1 = 5x4.

You can Read: Derivative of xn: Formula, Proof

Derivative of x^5 by First Principle

Derivative of x^5

The derivative of a function f(x) by the first principle is given by $\dfrac{d}{dx}$(f(x)) = limh→0 $\dfrac{f(x+h)-f(x)}{h}$.

Put f(x)=x5.

So the derivative of x5 will be equal to

$\dfrac{d}{dx}(x^5)$ = limh→0 $\dfrac{(x+h)^5-x^5}{h}$

[Let x+h=z, so that z→x when h→0. Note h=z-x]

So, $\dfrac{d}{dx}(x^5)$

= limz→x $\dfrac{z^5-x^5}{z-x}$

= 5x5-1 using the formula: limx→a $\dfrac{x^n-a^n}{x-a}$ = nan-1.

= 5x4.

So the derivative of x^5 is 5x4 and this is obtained by the first principle of derivative.

Also Read: Derivative of x4 by first principle

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Derivative of ecosx by first principle

Question-Answer

As an application, we now find the derivative of cosx to the power 5.

Question: What is the derivative of cos5x?

Answer:

Let z=cosx. So dz/dx = -sinx

By the chain rule of derivatives,

$\dfrac{d}{dx}(\cos^5 x)=\dfrac{d}{dx}(z^5)$

= $\dfrac{d}{dz}(z^5)$ \cdot \dfrac{dz}{dx}$

= $5z^4 \cdot (-\sin x)$

= – 5cos4x sinx as z=cosx.

So the derivative of cos^5x is equal to 4- 5cos4x sinx.

ALSO READ: Derivative of 1/(1+x)

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FAQs

Q1: What is the derivative of x5?

Answer: The derivative of x5 is equal to 5x4.

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