7.3 Differentiating powers, exponentials, trigonometric functions and logarithms — Learn

Prerequisites: the laws of indices for rational exponents; the derivative as a limit from Section 7.2; the graphs of $e^x$ and $\ln x$ from Chapter 6; the trigonometric functions from Chapter 5.

By the end of this section you can: differentiate $x^n$ for any rational $n$; differentiate a constant multiple, a sum and a difference of such terms; differentiate $e^{kx}$, $a^{kx}$, $\sin kx$, $\cos kx$, $\tan kx$ and $\ln x$; and choose the right rule for a given expression.

1. The power rule

For any rational number $n$, the derivative of $x^n$ is

Multiply by the old power, then reduce the power by one. The rule holds for whole numbers, fractions and negative numbers.

Example 1. Differentiate $y = x^4$.

Multiply by $4$ and reduce the power: $\dfrac{dy}{dx} = 4x^3$.

Example 2. Differentiate $y = \sqrt{x}$.

Write the root as a power: $y = x^{1/2}$. Then

Example 3. Differentiate $y = \dfrac{1}{x}$.

Write the fraction as a power: $y = x^{-1}$. Then

The derivative is negative everywhere, so the curve falls on each side of the break at $x = 0$.

A constant multiple, a sum and a difference are differentiated term by term:

Example 4. Differentiate $y = 4x^3 - 6x + \dfrac{1}{x}$.

Rewrite $\dfrac{1}{x}$ as $x^{-1}$ and differentiate each term:

Example 5. Differentiate $y = 2x^{3/2} + \dfrac{3}{x^2}$.

Rewrite $\dfrac{3}{x^2}$ as $3x^{-2}$. Then

2. Exponential functions

For a constant $k$,

The exponential function is special because its derivative has the same shape as the function itself. The constant $k$ appears in front. When $k = 1$ the derivative of $e^x$ is $e^x$.

For a positive base $a$,

The factor $\ln a$ appears because $a^{kx} = e^{kx\ln a}$.

Example 6. Differentiate $y = 5e^{3x} - 2e^{-x}$.

Example 7. Differentiate $y = 2^x$.

Here $a = 2$ and $k = 1$, so

3. Trigonometric functions

For a constant $k$,

The derivative of sine is cosine. The derivative of cosine is minus sine, so the minus sign often goes missing.

The tangent has its own result:

Example 8. Differentiate $y = 3\sin 2x - 4\cos x$.

Example 9. Differentiate $y = \tan 3x$.

4. The natural logarithm

For $x > 0$,

The rule also works for $\ln kx$ when $k > 0$, because $\ln kx = \ln k + \ln x$ and $\ln k$ is a constant. So

Example 10. Differentiate $y = 4\ln x - \ln 3x$.

The first term gives $\dfrac{4}{x}$, and the second gives $\dfrac{1}{x}$:

5. Try it: the power rule explorer

Predict first. The figure shows $y = x^n$ with a slider for $n$ and a point you can move. Before you touch the controls, predict the gradient at $x = 1$ for $n = 3$, and predict whether the gradient at $x = 1$ is larger or smaller when $n = 5$. Write both predictions down.

Test. Set $n = 3$ and move the point to $x = 1$. Read the gradient. Change $n$ to $5$ and read it again. Then set $n = 0.5$ and check the gradient at $x = 1$ and at $x = 4$.

Explain. Write two sentences. The first explains why the gradient at $x = 1$ equals $n$ for every curve $y = x^n$. The second explains why a curve with a negative power $n$ has a negative gradient.

Watch out. These mistakes are common.

Increasing the power instead of reducing it. The power rule is $x^n \to n x^{n-1}$. The power goes down by one. Writing $n x^{n+1}$ is a different function.

Forgetting the constant in front of an exponential. The derivative of $e^{3x}$ is $3e^{3x}$, not $e^{3x}$. The chain rule, met formally in Section 7.7, is what produces the $3$.

Dropping the minus sign on cosine. The derivative of $\cos kx$ is $-k\sin kx$. Check the sign with $k = 1$ at $x = 0$: the curve $y = \cos x$ is falling there, so its gradient must be negative.

Treating $\ln x$ as a power. The derivative of $\ln x$ is $\dfrac{1}{x}$, which is not $x^0$ or $0$. Re-check: differentiate $x$, $x^0$ and $\ln x$ separately and confirm that only the first gives $1$.

Progress check

Return to the box in the chapter problem. Its volume is $V = x(16 - 2x)(10 - 2x)$.

1. Expand the brackets to write $V$ as a cubic in $x$.
2. Differentiate $V$ with the power rule.
3. Setting $\dfrac{dV}{dx} = 0$ gives a quadratic. Write that quadratic down.

The next sections find the value of $x$ and prove that it gives the greatest volume.

You should now be able to

• differentiate $x^n$ for any rational $n$;
• differentiate a constant multiple, a sum and a difference;
• rewrite a root or a fraction as a power before differentiating;
• differentiate $e^{kx}$ and $a^{kx}$;
• differentiate $\sin kx$, $\cos kx$ and $\tan kx$, keeping the constant $k$ and the minus sign;
• differentiate $\ln x$ and $\ln kx$;
• choose the correct standard result for a given expression.