Derivative of Cosine
The derivative of cosine is negative sine.
Formula
Variables
Example
Slope of cos at 0 is 0
How it works
This rule tells you how fast cosine changes as the angle changes. It comes from the fact that cosine and sine are phase-shifted versions of the same circular motion: when you move a tiny amount along the unit circle, the instantaneous change in x-coordinate is given by the negative of the y-coordinate, which is why the derivative becomes -sin x. So wherever the cosine graph is rising or falling, the sine graph describes the slope, with a minus sign flipping the direction.
When to use it
Use this when you need the slope of a cosine graph at a point, or when differentiating expressions like cos(3x), cos(x^2), or more complicated formulas that contain cosine. It is common in physics, engineering, and signal processing whenever oscillations or circular motion are modeled with trig functions.
Worked example, step by step
Find the derivative of f(x) = 5cos x at x = \pi/3. Step 1: Differentiate the function: f'(x) = 5(-sin x) = -5sin x. Step 2: Substitute x = \pi/3: f'(\pi/3) = -5sin(\pi/3). Step 3: Use sin(\pi/3) = \sqrt{3}/2. Step 4: Compute the slope: f'(\pi/3) = -5(\sqrt{3}/2) = -5\sqrt{3}/2. Final answer: the slope of 5cos x at x = \pi/3 is -5\sqrt{3}/2.
Common mistakes to avoid
- Forgetting the negative sign and writing d/dx(cos x) = sin x
- Confusing the derivative of cos x with the derivative of sin x
- Treating x as degrees instead of radians when using the calculus rule
Did You Know?
Differentiating cosine gives negative sine, so a mass on a spring keeps trading position for velocity forever.
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