Derivative of Sine
The derivative of sine is cosine.
Formula
Variables
Example
Slope of sin at 0 is 1
How it works
This formula says that if you look at how fast sin x is changing at any angle x, that rate of change is cos x. It comes from the limit definition of a derivative: for very small changes in x, the vertical change in the sine graph behaves like the cosine value at that point. This is why the slope of the sine curve is positive where cosine is positive, zero where cosine is zero, and negative where cosine is negative.
When to use it
Use this when you need the slope of a sine graph, the rate of change of a periodic quantity, or the derivative of an expression containing sin x. It appears constantly in physics, engineering, and calculus problems involving waves, oscillations, and motion.
Worked example, step by step
Find the derivative of f(x) = sin x at x = \pi/3. Step 1: Use the rule \frac{d}{dx}\sin x = \cos x, so f'(x) = \cos x. Step 2: Substitute x = \pi/3: f'(\pi/3) = \cos(\pi/3). Step 3: Evaluate the cosine: \cos(\pi/3) = 1/2. Final answer: The slope of y = sin x at x = \pi/3 is 1/2.
Common mistakes to avoid
- Differentiating sin x as sin x instead of changing it to cos x
- Forgetting that the input angle x stays in radians when using calculus rules
- Mixing up the sign of the derivative in parts of the graph where cosine is negative
Did You Know?
The derivative of sine is cosine, meaning a sine wave and its rate of change are the same curve shifted a quarter turn.
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