1. Derivatives of the Sine, Cosine and Tangent Functions
by M. Bourne
Making use of an important result:
it can be shown from first principles that:
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:
Here is a demonstration of what the derivative of a transcendental curve really means, using LiveMath.
Example 1:
Find the derivative of y = sin(x2 + 3).
Here is the way LiveMath performs this derivative.
Normal 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.
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.
Here it is in LiveMath.
Normal answer:
4. Find the derivative of:
5. Find the derivative of y = 2x sin x + 2 cos x − x2cos 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
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
9. Show that y = cos3x tan x satisfies
10. Find the derivative of y = x tan x
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