Derivative of x^10: Proof by First Principle, Power Rule

The derivative of x^10 (x to the power 10) is 10x⁹. In this post, we will find the derivative of x¹⁰ by the first principle and the power rule of derivatives.

The derivative of x¹⁰ is denoted by d/dx (x¹⁰), and its formula is given by

$\dfrac{d}{dx}(x^{10})=10x^9$.

Derivative of x^10 by First Principle

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

Let us Put f(x)=x¹⁰.

Hence the derivative of x¹⁰ using the first principle is equal to

$\dfrac{d}{dx}(x^{10})$ = lim_h→0 $\dfrac{(x+h)^{10}-x^{10}}{h}$

= lim_h→0 $\dfrac{(x+h)^{10}-x^{10}}{(x+h)-x}$

[Let x+h=t, so that t→x when h→0]

Thus, $\dfrac{d}{dx}(x^{10})$

= lim_t→x $\dfrac{t^{10}-x^{10}}{t-x}$

= 10x^10-1 by the limit formula: lim_x→a $\dfrac{x^n-a^n}{x-a}$ = na^n-1.

= 10x⁹.

So the derivative of x^10 by the first principle is equal to 10x⁹.

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Derivative of x^10 by Power Rule

In the power rule of derivatives $\dfrac{d}{dx}(x^n)$ = nx^n-1, we put

n=10.

Thus, $\dfrac{d}{dx}(x^{10})$ = 10x^10-1 = 10x⁹.

So the derivative of x^10 by the power rule is 10x⁹.

Related Topics: Derivative of xⁿ: Formula, Proof

Question-Answer

We now find the derivative of e to the power 10x.

Question: Find the derivative of e^10x.

Answer:

Let z=e^x. So dz/dx = e^x.

By the chain rule of derivatives,

$\dfrac{d}{dx}(e^{10x})=\dfrac{d}{dx}(z^{10})$

= $\dfrac{d}{dz}(z^{10}) \cdot \dfrac{dz}{dx}$

= $10z^{10-1} \cdot e^x$

= 10 e^9x ⋅ e^x as z=e^x

= 10 e^10x.

So the derivative of e^{10x} is equal to 10 e^10x.

ALSO READ: Derivative of 1/ln(x)

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