1. Derivatives of the Sine, Cosine and Tangent Functions
by M. Bourne
It can be shown from first principles that:
`(d(sin\ x))/(dx)=cos\ x`
`(d(cos\ x))/dx=-sin\ x`
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:
`(d(sin\ u))/(dx)=cos\ u(du)/(dx)`
`(d(cos\ u))/dx=-sin\ u(du)/(dx)`
Differentiate `y = sin(x^2 + 3)`.
Find the derivative of `y = cos\ 3x^4`.
Differentiate `y = cos^3 2x`
Find the derivative of `y = 3\ sin\ 4x + 5\ cos\ 2x^3`.
1. Differentiate y = 4 cos (6x2 + 5).
2. Find the derivative of y = 3 sin3 (2x4 + 1).
3. Differentiate y = (x − cos2x)4.
4. Find the derivative of:
5. Differentiate 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
`y=(2\ sin\ 3x)/x`
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
`cos\ x(dy)/(dx)+3y\ sin\ x-cos^2x=0`
10. Find the derivative of y = x tan x
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