1.7.9 Differentiating x^n

What you'll learn

What a derivative tells you about a function.
How to differentiate $x^n$ when $n$ is any rational number.
How to differentiate constant multiples, sums and differences of powers.
How to handle roots, reciprocals and negative or fractional indices.

What does differentiation do?

The gradient of a straight line is constant. A curve, however, can have a different gradient at every point.

Differentiation is the process of finding a function that gives the gradient of a curve at any point. The resulting function is called the derivative.

For a function written as $y = f (x)$ , its derivative can be written as either

dydxorf′(x).\frac{dy}{dx} \qquad \text{or} \qquad f'(x).

Both notations mean “the derivative with respect to $x$ ”.

Definition

Derivative

The derivative $f^{'} (x)$ gives the gradient of the tangent to the curve $y = f (x)$ at each value of $x$ where the function is differentiable.

Once you have found the derivative, substituting a particular value of $x$ gives the gradient at that point.

The power rule

A term of the form $x^n$ is called a power of $x$ . The number $n$ is its index or exponent.

The main differentiation rule is:

ddx(xn)=nxn−1.\frac{d}{dx}\left(x^n\right)=nx^{n-1}.

To use the rule:

Multiply by the original power $n$ .
Reduce the power by 1.

Key Idea

The power rule

Bring the power down as a multiplier, then subtract 1 from the power:

xn⟶nxn−1.x^n \longrightarrow nx^{n-1}.

This rule applies when $n$ is rational, provided the original function and its derivative are defined at the value of $x$ being considered.

A rational number is a number that can be written as a fraction of two integers. This includes positive integers, negative integers, zero and fractions such as $12\frac{1}{2}$ or $−34-\frac{3}{4}$ .

Example

Differentiating a positive integer power

Differentiate $y=x^7$ .

Bring the power 7 down as a multiplier:
$dydx=7x□.\frac{dy}{dx}=7x^{\square}.$
Subtract 1 from the original power:
$7 - 1 = 6.$
Therefore,
$dydx=7x6.\frac{dy}{dx}=7x^6.$

Constant multiples

A constant is a quantity that does not depend on $x$ . In a term such as $5x^4$ , the number 5 is a constant multiplier.

When differentiating, the constant stays as a multiplier:

ddx(axn)=anxn−1,\frac{d}{dx}\left(ax^n\right)=anx^{n-1},

where $a$ is constant.

Example

Differentiating a constant multiple

Differentiate $y=-3x^5$ .

Apply the power rule to $x^5$ , bringing down the power 5:
$dydx=−3(5x5−1).\frac{dy}{dx}=-3\left(5x^{5-1}\right).$
Multiply the constants and simplify the power:
$\times 5=-15, \qquad 5-1=4.$
Hence,
$dydx=−15x4.\frac{dy}{dx}=-15x^4.$

Common Mistake

Forgetting the existing coefficient

When differentiating $ax^n$ , multiply the existing coefficient $a$ by the power $n$ . Do not replace $a$ with $n$ .

Sums, differences and constants

Differentiate each term in a sum or difference separately:

ddx(f(x)+g(x))=f′(x)+g′(x),\frac{d}{dx}\left(f(x)+g(x)\right)=f'(x)+g'(x),

and

ddx(f(x)−g(x))=f′(x)−g′(x).\frac{d}{dx}\left(f(x)-g(x)\right)=f'(x)-g'(x).

A standalone constant has derivative zero because its value never changes. Its graph is a horizontal line, which has gradient zero:

ddx(c)=0.\frac{d}{dx}(c)=0.

Example

Differentiating a polynomial

Differentiate

y=4x^5-3x^2+7x-6.

Differentiate $4x^5$ :
$4x5⟶4(5)x4=20x4.4x^5 \longrightarrow 4(5)x^4=20x^4.$
Differentiate the remaining variable terms, keeping their signs:
$−3x2⟶−3(2)x=−6x,-3x^2 \longrightarrow -3(2)x=-6x,$ $\longrightarrow 7.$
The constant term $- 6$ differentiates to zero, so
$dydx=20x4−6x+7.\frac{dy}{dx}=20x^4-6x+7.$

Tip

The derivative of a linear term

Since $x=x^1$ , the derivative of $a x$ is simply $a$ :

ddx(ax)=a.\frac{d}{dx}(ax)=a.

Negative powers

The power rule also works for negative rational indices.

For example,

x−3=1x3.x^{-3}=\frac{1}{x^3}.

Applying the power rule gives

ddx(x−3)=−3x−4=−3x4.\frac{d}{dx}\left(x^{-3}\right) =-3x^{-4} =-\frac{3}{x^4}.

It is usually easiest to rewrite reciprocals using negative indices before differentiating.

Example

Differentiating a reciprocal power

Differentiate

y=6x4.y=\frac{6}{x^4}.

Rewrite the denominator using a negative index:
$y=6x^{-4}.$
Apply the power rule:
$dydx=6(−4)x−4−1.\frac{dy}{dx}=6(-4)x^{-4-1}.$
Simplify, writing the answer either with a negative index or as a fraction:
$dydx=−24x−5=−24x5.\frac{dy}{dx}=-24x^{-5} =-\frac{24}{x^5}.$

Common Mistake

Changing the sign of a negative power

Subtracting 1 makes a negative power more negative. For example, $- 4 - 1 = - 5$ , not $- 3$ .

Fractional powers and roots

Roots can be written using fractional indices:

x=x1/2,x3=x1/3,x5=x5/2.\sqrt{x}=x^{1/2}, \qquad \sqrt[3]{x}=x^{1/3}, \qquad \sqrt{x^5}=x^{5/2}.

Rewriting roots in index form allows you to use the same power rule.

Example

Differentiating a square-root expression

Differentiate

y=8x−3x5.y=8\sqrt{x}-3\sqrt{x^5}.

Rewrite both roots as fractional powers:
$y=8x^{1/2}-3x^{5/2}.$
Apply the power rule to each term:
$dydx=8(12)x1/2−1−3(52)x5/2−1.\frac{dy}{dx} =8\left(\frac{1}{2}\right)x^{1/2-1} -3\left(\frac{5}{2}\right)x^{5/2-1}.$
Simplify the coefficients and indices:
$dydx=4x−1/2−152x3/2.\frac{dy}{dx} =4x^{-1/2}-\frac{15}{2}x^{3/2}.$
Equivalently,
$dydx=4x−152xx.\frac{dy}{dx} =\frac{4}{\sqrt{x}}-\frac{15}{2}x\sqrt{x}.$

Tip

Rewrite before differentiating

Convert roots and reciprocals into powers of $x$ first. This makes the power rule visible and reduces the chance of changing an index incorrectly.

Combining different types of power

An expression may contain positive, negative and fractional powers at the same time. The same rule applies independently to every term.

Example

Differentiating mixed rational powers

Given

f(x)=5x^{3/2}+2x^{-2}-4,

find $f^{'} (x)$ .

Differentiate the fractional-power term:
$5x3/2⟶5(32)x3/2−1=152x1/2.5x^{3/2} \longrightarrow 5\left(\frac{3}{2}\right)x^{3/2-1} =\frac{15}{2}x^{1/2}.$
Differentiate the negative-power term:
$2x−2⟶2(−2)x−3=−4x−3.2x^{-2} \longrightarrow 2(-2)x^{-3} =-4x^{-3}.$
The constant differentiates to zero, giving
$f′(x)=152x1/2−4x−3.f'(x)=\frac{15}{2}x^{1/2}-4x^{-3}.$

Finding a gradient at a point

The derivative is a function of $x$ . To find the gradient at a specific point, differentiate first and then substitute the given $x$ -coordinate.

Example

Finding the gradient at a point

The curve has equation $y=2x^{3/2}-5x$ . Find its gradient when $x = 4$ .

Differentiate each term:
$dydx=2(32)x1/2−5=3x1/2−5.\frac{dy}{dx} =2\left(\frac{3}{2}\right)x^{1/2}-5 =3x^{1/2}-5.$
Substitute $x = 4$ into the derivative:
$dydx∣x=4=34−5.\left.\frac{dy}{dx}\right|_{x=4} =3\sqrt{4}-5.$
Evaluate the exact value:
$3 (2) - 5 = 1.$
Therefore, the gradient is 1.

Common Mistake

Check the domain

Negative powers are undefined at $x = 0$ . Even roots such as $x\sqrt{x}$ are real only when $x≥0x\ge 0$ , and their derivatives may be undefined at an endpoint. Apply the power rule only where the relevant function is differentiable.

Exam technique

In the exam

Rewrite roots and reciprocals as fractional or negative powers before differentiating.
Differentiate every term separately, keeping each coefficient and sign attached to its term.
Check each differentiated power: the old power should become a multiplier, and the new power should be exactly 1 less.
Differentiate before substituting a given value of $x$ , and leave exact fractions or roots unless a decimal is requested.

Self review

Check yourself

Can you differentiate $7x^6-4x^3+9$ ?
Can you rewrite $3x\frac{3}{\sqrt{x}}$ as a power of $x$ and then differentiate it?
What is the gradient of $y=x^{3/2}+2x$ when $x = 9$ ?

What you'll learn

What a derivative tells you about a function.
How to differentiate $x^n$ when $n$ is any rational number.
How to differentiate constant multiples, sums and differences of powers.
How to handle roots, reciprocals and negative or fractional indices.

What does differentiation do?

The gradient of a straight line is constant. A curve, however, can have a different gradient at every point.

Differentiation is the process of finding a function that gives the gradient of a curve at any point. The resulting function is called the derivative.

For a function written as $y = f (x)$ , its derivative can be written as either

dydxorf′(x).\frac{dy}{dx} \qquad \text{or} \qquad f'(x).

Both notations mean “the derivative with respect to $x$ ”.

Definition

Derivative

The derivative $f^{'} (x)$ gives the gradient of the tangent to the curve $y = f (x)$ at each value of $x$ where the function is differentiable.

Once you have found the derivative, substituting a particular value of $x$ gives the gradient at that point.

The power rule

A term of the form $x^n$ is called a power of $x$ . The number $n$ is its index or exponent.

The main differentiation rule is:

ddx(xn)=nxn−1.\frac{d}{dx}\left(x^n\right)=nx^{n-1}.

To use the rule:

Multiply by the original power $n$ .
Reduce the power by 1.

Key Idea

The power rule

Bring the power down as a multiplier, then subtract 1 from the power:

xn⟶nxn−1.x^n \longrightarrow nx^{n-1}.

This rule applies when $n$ is rational, provided the original function and its derivative are defined at the value of $x$ being considered.

Example

Differentiating a positive integer power

Differentiate $y=x^7$ .

Bring the power 7 down as a multiplier:
$dydx=7x□.\frac{dy}{dx}=7x^{\square}.$
Subtract 1 from the original power:
$7 - 1 = 6.$
Therefore,
$dydx=7x6.\frac{dy}{dx}=7x^6.$

Constant multiples

A constant is a quantity that does not depend on $x$ . In a term such as $5x^4$ , the number 5 is a constant multiplier.

When differentiating, the constant stays as a multiplier:

ddx(axn)=anxn−1,\frac{d}{dx}\left(ax^n\right)=anx^{n-1},

where $a$ is constant.

Example

Differentiating a constant multiple

Differentiate $y=-3x^5$ .

Apply the power rule to $x^5$ , bringing down the power 5:
$dydx=−3(5x5−1).\frac{dy}{dx}=-3\left(5x^{5-1}\right).$
Multiply the constants and simplify the power:
$\times 5=-15, \qquad 5-1=4.$
Hence,
$dydx=−15x4.\frac{dy}{dx}=-15x^4.$

Common Mistake

Forgetting the existing coefficient

When differentiating $ax^n$ , multiply the existing coefficient $a$ by the power $n$ . Do not replace $a$ with $n$ .

Sums, differences and constants

Differentiate each term in a sum or difference separately:

ddx(f(x)+g(x))=f′(x)+g′(x),\frac{d}{dx}\left(f(x)+g(x)\right)=f'(x)+g'(x),

and

ddx(f(x)−g(x))=f′(x)−g′(x).\frac{d}{dx}\left(f(x)-g(x)\right)=f'(x)-g'(x).

A standalone constant has derivative zero because its value never changes. Its graph is a horizontal line, which has gradient zero:

ddx(c)=0.\frac{d}{dx}(c)=0.

Example

Differentiating a polynomial

Differentiate

y=4x^5-3x^2+7x-6.

Differentiate $4x^5$ :
$4x5⟶4(5)x4=20x4.4x^5 \longrightarrow 4(5)x^4=20x^4.$
Differentiate the remaining variable terms, keeping their signs:
$−3x2⟶−3(2)x=−6x,-3x^2 \longrightarrow -3(2)x=-6x,$ $\longrightarrow 7.$
The constant term $- 6$ differentiates to zero, so
$dydx=20x4−6x+7.\frac{dy}{dx}=20x^4-6x+7.$

Tip

The derivative of a linear term

Since $x=x^1$ , the derivative of $a x$ is simply $a$ :

ddx(ax)=a.\frac{d}{dx}(ax)=a.

Negative powers

The power rule also works for negative rational indices.

For example,

x−3=1x3.x^{-3}=\frac{1}{x^3}.

Applying the power rule gives

ddx(x−3)=−3x−4=−3x4.\frac{d}{dx}\left(x^{-3}\right) =-3x^{-4} =-\frac{3}{x^4}.

It is usually easiest to rewrite reciprocals using negative indices before differentiating.

Example

Differentiating a reciprocal power

Differentiate

y=6x4.y=\frac{6}{x^4}.

Rewrite the denominator using a negative index:
$y=6x^{-4}.$
Apply the power rule:
$dydx=6(−4)x−4−1.\frac{dy}{dx}=6(-4)x^{-4-1}.$
Simplify, writing the answer either with a negative index or as a fraction:
$dydx=−24x−5=−24x5.\frac{dy}{dx}=-24x^{-5} =-\frac{24}{x^5}.$

Common Mistake

Changing the sign of a negative power

Subtracting 1 makes a negative power more negative. For example, $- 4 - 1 = - 5$ , not $- 3$ .

Fractional powers and roots

Roots can be written using fractional indices:

x=x1/2,x3=x1/3,x5=x5/2.\sqrt{x}=x^{1/2}, \qquad \sqrt[3]{x}=x^{1/3}, \qquad \sqrt{x^5}=x^{5/2}.

Rewriting roots in index form allows you to use the same power rule.

Example

Differentiating a square-root expression

Differentiate

y=8x−3x5.y=8\sqrt{x}-3\sqrt{x^5}.

Rewrite both roots as fractional powers:
$y=8x^{1/2}-3x^{5/2}.$
Apply the power rule to each term:
$dydx=8(12)x1/2−1−3(52)x5/2−1.\frac{dy}{dx} =8\left(\frac{1}{2}\right)x^{1/2-1} -3\left(\frac{5}{2}\right)x^{5/2-1}.$
Simplify the coefficients and indices:
$dydx=4x−1/2−152x3/2.\frac{dy}{dx} =4x^{-1/2}-\frac{15}{2}x^{3/2}.$
Equivalently,
$dydx=4x−152xx.\frac{dy}{dx} =\frac{4}{\sqrt{x}}-\frac{15}{2}x\sqrt{x}.$

Tip

Rewrite before differentiating

Convert roots and reciprocals into powers of $x$ first. This makes the power rule visible and reduces the chance of changing an index incorrectly.

Combining different types of power

An expression may contain positive, negative and fractional powers at the same time. The same rule applies independently to every term.

Example

Differentiating mixed rational powers

Given

f(x)=5x^{3/2}+2x^{-2}-4,

find $f^{'} (x)$ .

Differentiate the fractional-power term:
$5x3/2⟶5(32)x3/2−1=152x1/2.5x^{3/2} \longrightarrow 5\left(\frac{3}{2}\right)x^{3/2-1} =\frac{15}{2}x^{1/2}.$
Differentiate the negative-power term:
$2x−2⟶2(−2)x−3=−4x−3.2x^{-2} \longrightarrow 2(-2)x^{-3} =-4x^{-3}.$
The constant differentiates to zero, giving
$f′(x)=152x1/2−4x−3.f'(x)=\frac{15}{2}x^{1/2}-4x^{-3}.$

Finding a gradient at a point

The derivative is a function of $x$ . To find the gradient at a specific point, differentiate first and then substitute the given $x$ -coordinate.

Example

Finding the gradient at a point

The curve has equation $y=2x^{3/2}-5x$ . Find its gradient when $x = 4$ .

Differentiate each term:
$dydx=2(32)x1/2−5=3x1/2−5.\frac{dy}{dx} =2\left(\frac{3}{2}\right)x^{1/2}-5 =3x^{1/2}-5.$
Substitute $x = 4$ into the derivative:
$dydx∣x=4=34−5.\left.\frac{dy}{dx}\right|_{x=4} =3\sqrt{4}-5.$
Evaluate the exact value:
$3 (2) - 5 = 1.$
Therefore, the gradient is 1.

Common Mistake

Check the domain

Exam technique

In the exam

Rewrite roots and reciprocals as fractional or negative powers before differentiating.
Differentiate every term separately, keeping each coefficient and sign attached to its term.
Check each differentiated power: the old power should become a multiplier, and the new power should be exactly 1 less.
Differentiate before substituting a given value of $x$ , and leave exact fractions or roots unless a decimal is requested.

Self review

Check yourself

Can you differentiate $7x^6-4x^3+9$ ?
Can you rewrite $3x\frac{3}{\sqrt{x}}$ as a power of $x$ and then differentiate it?
What is the gradient of $y=x^{3/2}+2x$ when $x = 9$ ?

What you'll learn

What does differentiation do?

The power rule

Constant multiples

Sums, differences and constants

Negative powers

Fractional powers and roots

Combining different types of power

Finding a gradient at a point

Review 1.7.9 Differentiating x^n by teaching Genie

1.7.9 Differentiating x^n

What you'll learn

What does differentiation do?

The power rule

Constant multiples

Sums, differences and constants

Negative powers

Fractional powers and roots

Combining different types of power

Finding a gradient at a point

Review 1.7.9 Differentiating x^n by teaching Genie

1.7.9 Differentiating x^n

What you'll learn

What does differentiation do?

The power rule

Constant multiples

Sums, differences and constants

Negative powers

Fractional powers and roots

Combining different types of power

Finding a gradient at a point

Review 1.7.9 Differentiating x^n by teaching Genie

1.7.9 Differentiating x^n Revision Guide

1.7.9 Differentiating x^n

What you'll learn

What does differentiation do?

The power rule

Constant multiples

Sums, differences and constants

Negative powers

Fractional powers and roots

Combining different types of power

Finding a gradient at a point

Review 1.7.9 Differentiating x^n by teaching Genie

1.7.9 Differentiating x^n Revision Guide