If g is a differentiable function find an expression for

Question

1) Suppose f(π/4) = ½ and f'(π/4) = -3, and let g(x) = f(x)sin(x) and h(x) = cos(x)/f(x). Find the following derivatives:

a. g'(π/4)

b. h'(π/4)

2) If f(3) = 4, g(3) = 2, f'(x) = -6 and g'(x) = 5, find the following numbers.

a. (f+g)'(3)

b. (fg)'(3)

c. (f/g)'(3)

d. (g/f)'(3)

3) Let P(x) = F(x)G(x) and Q(x) = F(x)/G(x), where F and G are the following whose graphs are shown.

a. Find P'(2)

b. Find Q'(7)

4) If g is a differentiable function, find an expression for the derivative of each of the following functions

a. y = x³g(x)

b. y = g(x)/x⁴

 c. y = x²/g(x)

d. y = 1+xg(x)/ √x

5) If h(x) = √(3+2f(x)), where f(2) = 3 and f'(2) = 5, find h'(2).

6) A table of values for f, g, f' and g' is given.

x	f(x)	g(x)	f'(x)	g'(x)
1	3	2	4	6
2	1	8	5	7
3	7	2	7	9

a. If F(x) = f(f(x)), find F'(2).

b. If G(x) = g(g(x)), find G'(3).

c. If H(x) = g(f(x)), find H'(2).

7) If f is the function whose graph is shown, let h(x) = f(f(x)) and g(x) = f(x²). Use the graph of f to estimate the value of each derivative.

a. h'(2)

b. g'(2)

8) Suppose f is differentiable on R and α is a real number. Let F(x) = f(x^α)  and G(x) = [f(x)]^α. Find expressions for:

a. F'(x)

b. G'(x)

9) If g is a twice differentiable function and f(x) = g(x³) sin(x), find f" in terms of g, g', and g".