Derivative of Sine
Derivative of the sine function.
Formula
Variables
Example
The slope of sin x at x = 0 is cos 0 = 1.
How it works
This rule says that the slope of the graph y = sin x at any point x is given by cos x. It comes from the limit definition of the derivative: for very small changes in x, the change in sin x is controlled by the angle addition identity, and after simplifying, the limit becomes cos x. The result means the sine curve rises most steeply where cosine is 1 and flattens where cosine is 0.
When to use it
Use this when you need the instantaneous rate of change of a sine function, such as in motion, oscillations, waves, or any model involving periodic behavior. It is also the first step in differentiating more complicated expressions that contain sin x.
Worked example, step by step
Find the derivative of y = sin x and evaluate it at x = π/3. Step 1: Differentiate using the rule d/dx(sin x) = cos x. So, y' = cos x. Step 2: Substitute x = π/3. y'(π/3) = cos(π/3) Step 3: Use the exact trig value. cos(π/3) = 1/2 Final answer: the slope at x = π/3 is 1/2.
Common mistakes to avoid
- Forgetting that the derivative of sin x is cos x, not sin x
- Using degrees instead of radians when applying the rule in calculus
- Confusing the derivative of sin x with the derivative of sin(ax), which needs the chain rule
Did You Know?
The derivative of sine is cosine — differentiate four times and you cycle right back to sine.
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