1. If f(x)=sin(x^2), Find f'(x) and f'(1)

Question

2. If d/dx(f(3x^3))=9x^2, Calculate f'(x).
3. Let f(x)=cos(6x+6), find f'(x)
4. Let f(x)=-2sec(2x), find f'(x)

Answer 1

Looks like an exercise in the chain rule:

f(x) = sin(x^2)
f'(x) = cos(x^2) * 2x = 2x cos(x^2)
f'(1) = 2cos(1)
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d/dx(f(3x^3))=9x^2

Let g(x) = f(3x^3)
dg/dx = dg/df * df/dx
9x^2 = dg/df * 9x^2
1 = dg/df
Looks like g(x) = x
so, g(f) = f
dg/dx = df/dx = 9x^2
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f(x)=cos(6x+6)
f' = -sin(6x+6)*6
= -6sin(6x+6)
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f(x)=-2sec(2x)
f' = -2sec(2x)tan(2x)*2
= -2sec(2x)tan(2x)