Derivatives of sine, cosine and tangent

What you will be able to do

• derive the derivatives of $\sin x$, $\cos x$ and $\tan x$ from the definition;
• use the results to find slopes and horizontal tangents;
• state the derivatives of $e^x$ and $\ln x$ and where they come from.

Why sine needs the standard limits

The derivative of $x^n$ came from a factorisation. Sine has no such factorisation. Instead we use the addition formula and the standard limits of the first course. All angles are in radians.

The derivative of sine

Let $f(x) = \sin x$. Then

Split this into two parts:

As $h \to 0$ we have $\dfrac{\sin h}{h} \to 1$. Also $\dfrac{1 - \cos h}{h} = h \cdot \dfrac{1 - \cos h}{h^2} \to 0 \times \dfrac{1}{2} = 0$. So

The picture shows why this is reasonable. Where the sine curve rises fastest, at $x = 0$, the slope is $1$, which is $\cos 0$. At the top of the wave, $x = \dfrac{\pi}{2}$, the tangent is flat, and $\cos\dfrac{\pi}{2} = 0$.

The derivative of cosine

In the same way,

As $h \to 0$ the first part tends to $0$ and the second tends to $-\sin x$.

So $\dfrac{d}{dx}\cos x = -\sin x$. The minus sign is the one to remember: just after $x = 0$ the cosine curve falls.

The derivative of tangent

For $f(x) = \tan x$ use $\tan(x + h) - \tan x = \dfrac{\sin\big((x + h) - x\big)}{\cos(x + h)\cos x} = \dfrac{\sin h}{\cos(x + h)\cos x}$. Then

As $h \to 0$ this tends to $\dfrac{1}{\cos^2 x} = \sec^2 x$.

So $\dfrac{d}{dx}\tan x = \sec^2 x$. It is always positive: the tangent curve is rising wherever it is defined.

Exponential and logarithm

The standard limits of the first course give two more results at once.

• $\dfrac{e^{x + h} - e^x}{h} = e^x \cdot \dfrac{e^h - 1}{h} \to e^x$. So $\dfrac{d}{dx}e^x = e^x$, the function that equals its own derivative.
• $\dfrac{\ln(x + h) - \ln x}{h} = \dfrac{\ln\left(1 + \frac{h}{x}\right)}{h} = \dfrac{1}{x} \cdot \dfrac{\ln\left(1 + \frac{h}{x}\right)}{h/x} \to \dfrac{1}{x}$. So $\dfrac{d}{dx}\ln x = \dfrac{1}{x}$ for $x > 0$.

Also $\dfrac{d}{dx}a^x = a^x\ln a$, because $\dfrac{a^h - 1}{h} \to \ln a$.

A table to keep

The derivatives of $\sec x$, $\csc x$ and $\cot x$ come from the quotient rule in the next lesson.

A worked example

Find where the tangent to $y = \sin x$ has slope $\dfrac{1}{2}$ for $0 < x < \dfrac{\pi}{2}$.

1. The slope is $\cos x$. We need $\cos x = \dfrac{1}{2}$.
2. In the interval, $\cos x = \dfrac{1}{2}$ at $x = \dfrac{\pi}{3}$.
3. The point is $\left(\dfrac{\pi}{3}, \dfrac{\sqrt{3}}{2}\right)$.

Another: the tangent to $y = \cos x$ is horizontal where $-\sin x = 0$, which is at $x = 0, \pm\pi, \pm 2\pi, \dots$

Practise with new numbers

Try it in Python

The program compares the difference quotient with the formula for $\sin x$, $\cos x$ and $\tan x$ at $x = 0.7$. It also prints $\dfrac{1 - \cos h}{h}$ for smaller and smaller $h$.

Common mistakes

• Degrees. The formulas $\dfrac{d}{dx}\sin x = \cos x$ and so on are true for $x$ in radians only.
• Losing the minus sign for cosine. $\dfrac{d}{dx}\cos x = -\sin x$.
• Writing $\dfrac{d}{dx}\tan x = \sec x$. The square is part of the answer: $\sec^2 x$.
• Confusing $\dfrac{d}{dx}e^x$ with $xe^{x-1}$. The power rule does not apply to $e^x$, because the variable is in the exponent.

Remember these

• $(\sin x)' = \cos x$, $(\cos x)' = -\sin x$, $(\tan x)' = \sec^2 x$, with $x$ in radians.
• $(e^x)' = e^x$, $(\ln x)' = \dfrac{1}{x}$, $(a^x)' = a^x\ln a$.
• The proofs use $\dfrac{\sin h}{h} \to 1$ and $\dfrac{1 - \cos h}{h} \to 0$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find the slope of the tangent to $y = \sin x$ at $x = \dfrac{\pi}{3}$.

2. Find the slope of $y = \tan x$ at $x = \dfrac{\pi}{4}$.

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

4. At which values of $x$ in $[0, 2\pi]$ is the tangent to $y = \sin x$ horizontal?

5. Why does the proof for $\sin x$ need $\dfrac{1 - \cos h}{h} \to 0$?

6. Find the slope of $y = \ln x$ at $x = 4$.

References

• OpenStax, Calculus Volume 1, Chapter 3, section 3.5 on the derivatives of trigonometric functions.
• NCERT, Mathematics Textbook for Class XI, Chapter 12, the derivatives of $\sin x$ and $\cos x$ from the definition.