Key takeaways
- d/dx sin(x) = cos(x), and d/dx sin(ax) = a·cos(ax).
- The extra a comes from the chain rule: the inside ax has derivative a.
- The derivative of sin is +cos. The minus sign belongs to the derivative of cos.
- x must be in radians, not degrees.
In this guide
What is the derivative of sin?
A derivative is the slope of a graph at one point: how steep it is and whether it is going up or down. The sine graph is a wave. At x = 0 it rises with slope 1. At the top of the wave (x = π/2) it is flat, so the slope is 0. Going down the other side, the slope becomes negative.
If you plot all those slopes, you get another wave that starts at 1: the cosine wave. That is why d/dx sin x = cos x.
Why does the derivative of sin matter?
Anything that goes up and down in a regular rhythm can be described with sine: a swing, a spring, a guitar string, the voltage in a wall socket. If sine gives the position, its derivative gives the speed.
The rule only looks this clean in radians. In degrees, the derivative of sin x would be (π/180)·cos x, about 0.01745·cos x. That tiny messy number is why mathematicians use radians.
The formula
| Symbol | What it means | Example |
|---|---|---|
| a | The number multiplying x inside the sine. Bigger a = faster wave | 2 |
| x | The point where you want the slope, in radians | 0.5 |
| cos(ax) | Cosine of the same angle | cos(1) ≈ 0.5403 |
| a·cos(ax) | The slope, which the calculator works out | 2 × 0.5403 ≈ 1.0806 |
The extra a comes from the chain rule: a faster wave (bigger a) is also steeper, so its slope is a times bigger.
How to differentiate sin(ax) step by step
- 1Find a. It is the number in front of x inside the sine. For sin(x), a = 1.
- 2Change sin to cos. Keep the same inside: cos(ax).
- 3Multiply by a. You now have a·cos(ax).
- 4Put in x for a slope at a point. Use radians: work out ax, find its cosine, multiply by a.
Now try it yourself. Enter a and x (in radians) in the calculator below to see the slope with every step.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Forgetting the a.
d/dx sin(2x)is2cos(2x), notcos(2x). - Dividing by a. Dividing is for integrals:
∫cos(ax) dx = sin(ax)/a + C. Derivatives multiply. - Using degrees. Convert first with the degrees to radians calculator: 90° = π/2 ≈ 1.5708.
Tips and tricks
- Numbers in front stay:
d/dx 5sin(x) = 5cos(x). - Other insides: for
sin(x²), the inside derivative is 2x, so the answer is2x·cos(x²). - Squares:
sin²x = (sin x)², so its derivative is2 sin x cos x, which also equalssin(2x).
Where you'll use it in real life
- Springs and pendulums: if height is
sin(ωt), the speed isω·cos(ωt). - Sound and music: a pure note is a sine wave. Its slope tells you how fast the air pressure changes.
- Electricity: AC voltage is a sine wave, and the current through some parts depends on its rate of change.
- Tides and seasons: things that repeat every day or year are often modelled with sine. Look up sine values with the sin cos tan calculator.
Quick summary
The derivative of sin(ax) is a·cos(ax): switch sin to cos, keep the inside, and multiply by a. Work in radians and don’t add a minus sign. Use the calculator above to find the slope at any point, with every step shown.
Worked example
A weight on a spring moves up and down, and its height is y = sin(2t) metres after t seconds. How fast is it moving (in metres per second) at t = 0.5 seconds? The number in front of t is a = 2, and x = 0.5.
- 01d/dx[sin(2x)] = 2·cos(2x)
- 02At x = 0.5: 2 · cos(1 rad)
- 03cos(1) = 0.5403
- 04= 2 × 0.5403 = 1.0806
These numbers are pre-filled in the calculator above.
Frequently asked questions
What is the derivative of sin 2x?
It is 2cos(2x). Change sin to cos, keep the 2x inside, and multiply by 2.
Why do I need radians?
The neat result d/dx sin x = cos x is only true when x is in radians. In degrees, you would get an extra factor of π/180.
What is the derivative of sin²x?
Treat it as (sin x)² and use the chain rule: 2 sin x × cos x, which also equals sin(2x).
