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

Here is a demonstration of what the derivative of a transcendental curve really means, using LiveMath.

LIVEMath

 

Example 1:

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


Here is the way LiveMath performs this derivative.

LIVEMath


Normal answer:

Answer


IMPORTANT:

cos x2 + 3 ≠ cos(x2 + 3).

Can you see why they must be different?


Example 2:

Find the derivative of y = cos 3x4.


Answer

 

Example 3:

Find the derivative of y = cos32x


Answer

 

Example 4:

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


Answer

 

Exercises:

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


Answer


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


Answer


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


Here it is in LiveMath.

LIVEMath


Normal answer:

Answer


4. Find the derivative of:

math expression

Answer


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


Answer


6. Find the derivative of the implicit function

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


Answer


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

math expression where x = 0.15.


Answer


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

Answer


9. Show that y = cos3x tan x satisfies

math expression

Answer


10. Find the derivative of y = x tan x


Answer




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