Derivative of trigonometric functions

Example 1

Find the derivative functions of the functions below

Derivatives

Example 2

Find the value of the derivative in 𝜋

The function consists of the product of two functions. The product derivation rule is used

We substitute 𝜋

Example 3

Find the zeros of the derivative function

Derivative

Zeros

where n is an integer.

Example 4

Find the extreme values of the function

Differentiate the function. The sine derivative is cosine and the cosine derivative is minus sine. In addition, in a function, a term with a cosine square is a combined function.

Zeros

cosine is 0 when

Sine is 1 when

The derivative is always positive before the zeros determined by the angle 𝜋 / 2 and negative thereafter before the zeros determined by 3𝜋 / 2

g '(0) = 2

g '(π) = - 2

So at 𝜋 / 2 + n2𝜋 the function has maximum points and at 3𝜋 / 2 + n2𝜋 the function has minimum points.

The extreme values

In the figure, the graph of the function g(x) in green and the graph of the derivative g'(x) in red.

Definition of tangent

Then the derivative of the tangent is obtained by the derivation rule of the quotient

The derivative of the tangent is not defined at cosine zeros. Just like the tangent.

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