The derivative of cosine x is negative sine x. This is a fundamental result in calculus. When we graph both the original cosine function in blue and its derivative negative sine x in red, we can see how the derivative represents the slope of the tangent line at each point on the cosine curve.
The derivative has a clear geometric meaning. At any point on the cosine curve, the derivative gives us the slope of the tangent line. As we move along the curve, we can see how the slope changes. When cosine is decreasing, the slope is negative, and when cosine is increasing, the slope is positive.
On the unit circle, any point can be represented as cosine theta, sine theta. As the point moves around the circle, the x-coordinate changes at a rate proportional to negative sine theta. The green arrow shows the velocity vector, which points in the direction tangent to the circle.
When dealing with composite functions, we apply the chain rule. For example, the derivative of cosine of 2x equals negative sine of 2x times the derivative of 2x, which is 2. This gives us negative 2 sine of 2x. The graph shows how the derivative has twice the frequency and amplitude.
To summarize, the derivative of cosine x is negative sine x. This fundamental result has important geometric meaning as the slope of the tangent line. When combined with the chain rule, we can differentiate more complex functions. This derivative formula is essential in physics for describing oscillatory motion, in engineering for signal analysis, and in mathematics for solving optimization problems.