Derivatives of the Trigonometric Functions

Consider a Ferris wheel rotating at a constant speed. The vertical height of a point on its edge changes smoothly over time, following a sine wave.

The rider's rate of change of height also varies smoothly and cyclically, like another wave. This changing rate matches the derivative of the sine function.

To find the derivative of sine x, start from the limit definition of a derivative and apply it to sine x.

First, expand the sine of (x + h) using the addition formula. Then, rearrange the terms and factor out sine x and cosine x.

As h approaches zero, cosine h minus 1 over h approaches zero, and sine h over h approaches 1.

Substituting these limits shows that the derivative of sine x is cosine x.

The same approach shows that the derivative of cosine x is negative sine x.

Since the tangent function is the ratio of sine to cosine, to find the derivative, apply the quotient rule. Now, substituting the values of the derivatives of the sine and cosine functions and simplifying using trigonometric identities gives the derivative of the tangent as secant squared.