Differentiating $$x^n$$​​

In a nutshell

The general rule that can be applied when differentiating $$y=x^n$$ with respect to $$x$$ is to "multiply by the power $$n$$, then subtract $$1$$ from the power". This rule is valid for any $$n$$.

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.

$$y$$ notation

Sometimes equations are presented in a form like $$y=x^n$$. When this is differentiated with respect to $$x$$, the $$y$$ part becomes $$\dfrac{\text{d}y}{\text{d}x}$$. This is because to "differentiate $$y$$ with respect to $$x$$" is in essence to apply $$\dfrac{\text{d}}{\text{d}x}$$ to $$y$$.

Function notation

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

The derivative of $$x^n$$​​

If $$y=x^n$$ where $$n$$ is a constant number, then the derivative will be:

​$$\boxed{\dfrac{\text{d}y}{\text{d}x}=nx^{n-1}}$$​​

See that the expression has been multiplied by the power $$n$$, then $$1$$ was subtracted from the power.

If $$y=ax^n$$ where $$a$$ is also a constant, then the derivative will be:

​$$\boxed{\dfrac{\text{d}y}{\text{d}x}=anx^{n-1}}$$​​

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

Using the function notation, if $$f(x)=x^n$$, then the derivative will be:

$$\boxed{f'(x)=nx^{n-1}}$$​​

If $$f(x)=ax^n$$, then the derivative will be:

​$$\boxed{f'(x)=anx^{n-1}}$$​​

Example 1

Find the derivative of $$y=x^7$$.

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

$$\underline{\dfrac{\text{d}y}{{\text{d}x}}=7x^6}$$​​

Example 2

Find the derivative of $$f(x)=3x^2$$.

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

$$\underline{f'(x)=6x}$$​​

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Example 3

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

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

$$\underline{\dfrac{\text{d}y}{\text{d}x}=\frac32x^{\frac12}}$$​​

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Example 4

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

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

$$\underline{f'(x)=-45x^{-10}}$$​​

Example 5

Find the derivative of $$f(x)=\frac3{\sqrt{x}}$$.

First express this in terms of $$f(x)=ax^n$$:

$$f(x)=\frac3{\sqrt{x}}\\\space\\f(x)=\frac3{x^{\frac12}}\\\space\\f(x)=3x^{-\frac12}$$​​

Now apply the rule for differentiation:

$$\underline{f'(x)=-\dfrac 3 2 x ^{-\frac 3 2}}$$​​