Derivative of x^6: Proof by First Principle, Power Rule

The derivative of x^6 (x to the power 6) is equal to 6x5. Here, we will find the derivative of x6 using the power rule and the first principle of derivatives.

The derivative of x6 is denoted by d/dx (x6). The derivative formula of x6 is as follows:

$\dfrac{d}{dx}(x^6)=6x^5$.

Derivative of x^6 by Power Rule

By the power rule of derivatives $\dfrac{d}{dx}(x^n)$ = nxn-1, we get that

$\dfrac{d}{dx}(x^6)$ = 6x6-1 = 6x5.

Hence the derivative of x6 by the power rule is 6x5.

You can Read: Derivative of xn: Formula, Proof

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Derivative of x^6 by First Principle

Derivative of x^6 by First Principle

In first principle of derivatives formula $\dfrac{d}{dx}$(f(x)) = limh→0 $\dfrac{f(x+h)-f(x)}{h}$, we put

f(x)=x6.

So the derivative of x6 by first principle is given by

$\dfrac{d}{dx}(x^6)$ = limh→0 $\dfrac{(x+h)^6-x^6}{h}$

= limh→0 $\dfrac{(x+h)^6-x^6}{(x+h)-x}$

[Let x+h=z, so that z→x when h→0]

Thus, $\dfrac{d}{dx}(x^6)$

= limz→x $\dfrac{z^6-x^6}{z-x}$

= 6x6-1 obtained by the formula limx→a $\dfrac{x^n-a^n}{x-a}$ = nan-1.

= 6x5.

So the derivative of x^6 is 6x5 and this is proved by the first principle.

Also Read: Derivative of 2x by first principle

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FAQs

Q1: What is the derivative of x6?

Answer: As d/dx (xn) = nxn-1, the derivative of x6 is equal to 6x5.

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