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Differentiation I

Differentiating x^n

Differentiating x^n

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Summary

Differentiating xnx^nxn​​

In a nutshell

The general rule that can be applied when differentiating y=xny=x^ny=xn with respect to xxx is to "multiply by the power nnn, then subtract 111 from the power". This rule is valid for any nnn.



Notation

When differentiating, there are two common forms of notation. The rules of differentiation apply to both in the same way, they just offer different ways to express equations. The context in which they're used determines which notation is most suitable.


yyy notation

Sometimes equations are presented in a form like y=xny=x^ny=xn. When this is differentiated with respect to xxx, the yyy part becomes dydx\dfrac{\text{d}y}{\text{d}x}dxdy​. This is because to "differentiate yyy with respect to xxx" is in essence to apply ddx\dfrac{\text{d}}{\text{d}x}dxd​ to yyy.


Function notation

Alternatively, if the equation is given as f(x)=xnf(x)=x^nf(x)=xn. When this is differentiated with respect to xxx, the f(x)f(x)f(x)​ part becomes f′(x)f'(x)f′(x). This signifies it has been differentiated once. If differentiated more times, there would be more dashes.



The derivative of xnx^nxn​​

If y=xny=x^ny=xn where nnn is a constant number, then the derivative will be:

​dydx=nxn−1\boxed{\dfrac{\text{d}y}{\text{d}x}=nx^{n-1}}dxdy​=nxn−1​​​


See that the expression has been multiplied by the power nnn, then 111 was subtracted from the power.


If y=axny=ax^ny=axn where aaa is also a constant, then the derivative will be:

​dydx=anxn−1\boxed{\dfrac{\text{d}y}{\text{d}x}=anx^{n-1}}dxdy​=anxn−1​​​


This is the same as above, but there is a multiple aaa involved.


Using the function notation, if f(x)=xnf(x)=x^nf(x)=xn, then the derivative will be:

f′(x)=nxn−1\boxed{f'(x)=nx^{n-1}}f′(x)=nxn−1​​​


If f(x)=axnf(x)=ax^nf(x)=axn, then the derivative will be:

​f′(x)=anxn−1\boxed{f'(x)=anx^{n-1}}f′(x)=anxn−1​​​


Example 1

Find the derivative of y=x7y=x^7y=x7.


The rule is to multiply by the power, then subtract one from the power:

dydx=7x6‾\underline{\dfrac{\text{d}y}{{\text{d}x}}=7x^6}dxdy​=7x6​​​


Example 2

Find the derivative of f(x)=3x2f(x)=3x^2f(x)=3x2.


The rule is to multiply by the power, then subtract one from the power:

f′(x)=6x‾\underline{f'(x)=6x}f′(x)=6x​​​

​

Example 3

​Find the derivative of y=x32y=x^{\frac32}y=x23​.


The rule is to multiply by the power, then subtract one from the power:

dydx=32x12‾\underline{\dfrac{\text{d}y}{\text{d}x}=\frac32x^{\frac12}}dxdy​=23​x21​​​​

​

Example 4

Find the derivative of f(x)=5x−9f(x)=5x^{-9}f(x)=5x−9.


The rule is to multiply by the power, then subtract one from the power:

f′(x)=−45x−10‾\underline{f'(x)=-45x^{-10}}f′(x)=−45x−10​​​


Example 5

Find the derivative of f(x)=3xf(x)=\frac3{\sqrt{x}}f(x)=x​3​.


First express this in terms of f(x)=axnf(x)=ax^nf(x)=axn:

f(x)=3x f(x)=3x12 f(x)=3x−12f(x)=\frac3{\sqrt{x}}\\\space\\f(x)=\frac3{x^{\frac12}}\\\space\\f(x)=3x^{-\frac12}f(x)=x​3​ f(x)=x21​3​ f(x)=3x−21​​​


Now apply the rule for differentiation:

f′(x)=−32x−32‾\underline{f'(x)=-\dfrac 3 2 x ^{-\frac 3 2}}f′(x)=−23​x−23​​​​


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FAQs - Frequently Asked Questions

What does f(x) become when it is differentiated with respect to x?

It become f'(x).

What does y become when differentiated with respect to x?

It becomes dy/dx.

How do you differentiate x^n?

Multiply by the power n, then subtract 1 from the power.

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