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:

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

Put n=5.

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

You can Read: Derivative of xⁿ: Formula, Proof

Derivative of x^5 by First Principle

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

Put f(x)=x⁵.

So the derivative of x⁵ will be equal to

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

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

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

= lim_z→x $\dfrac{z^5-x^5}{z-x}$

= 5x^5-1 using the formula: lim_x→a $\dfrac{x^n-a^n}{x-a}$ = na^n-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

Derivative of e^sinx by first principle

Derivative of e^cosx 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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