1. Derivatives of the Sine, Cosine and Tangent Functions

by M. Bourne

It can be shown from first principles that:

math expression

In words, we would say:

The derivative of sin x is cos x,
The derivative of cos x is −sin x (note the negative sign!) and
The derivative of tan x is sec2x.

Now, if u = f(x) is a function of x, then by using the chain rule, we have:

math expression

Example 1:

Find the derivative of y = sin(x2 + 3).

Example 2:

Find the derivative of y = cos 3x4.

Example 3:

Find the derivative of y = cos32x

Example 4:

Find the derivative of y = 3 sin 4x + 5 cos 2x3.

Exercises:

1. Find the derivative of y = 4 cos (6x2 + 5).

2. Find the derivative of y = 3 sin3 (2x4 + 1).

3. Find the derivative of y = (x − cos2x)4.

4. Find the derivative of:

math expression

5. Find the derivative of y = 2x sin x + 2 cos xx2cos x.

6. Find the derivative of the implicit function

x cos 2y + sin x cos y = 1.

7. Find the slope of the line tangent to the curve of

math expression

where x = 0.15

8. The current (in amperes) in an amplifier circuit, as a function of the time t (in seconds) is given by

i = 0.10 cos (120πt + π/6).

Find the expression for the voltage across a 2.0 mH inductor in the circuit, given that

math expression

9. Show that y = cos3x tan x satisfies

math expression

10. Find the derivative of y = x tan x

See also: Derivative of square root of sine x by first principles.

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