Find C1, C2, and Ca so that the given functions will satisfy the prescribed initial conditions. 4a. y(x)=C,e" +C,e* + 3e* ; y(0)3Dy'(0)%=D0 4b. y(x)=C,e" +C,xe" +C,x'e* ; y'(1) =e; y'(1)= 2e ; "(1)=5e

Find y as a function of a if y(0) = 7, y'(0) = 4, y"(0) = 133. (1) = ')} y" - 8y" -y + 8y = 0,

6. If g(x) = x'+x²+ g"(2). find the value of

5. If y = 5x²-3x + 10, give y'(-1)

The graphs of f and g are shown. Let h(x) = f(g(x)) and s(x) = g(f(x)). y 15 10 5 -2 4 6. 10 (a) Write an equation for h'(x) in terms of f(x), g(x), f'(x), and g'(x). O h'(x) = g'(f'(x))f'(x) O h'(x) = g'(f(x))f'(x) h'(x) = f'(g'(x))g'(x) h'(x) = f'(g(x))gʻ(x) O h'(x) = f'(g'(x))g(x) O h'(x) = g'(f'(x))f(x) Find h'(3). (If an answer does not exist, enter DNE.) (b) Write an equation for s'(x) in terms of f(x), g(x), f'(x), and g'(x). O s'(x) = f'(g(x))g'(x) O s'(x) = g'(f'(x))F'(x) O s'(x) = f'(g'(x))g'(x) s'(x) = g'(f(x))f'(x) s'(x) = f'(g'(x))g(x) s'(x) = g'(f'(x)){x) Find s'(7). (If an answer does not exist, enter DNE.)

2.Given F(x)=h(f(g(x))), g(1) = 2, g'(1) = 3, f(2) = 4, h'(4) = 1 and f'(2) = 6, findF'(1)

1. let y = x(x2+x)2  , Find y'   2. let f(x) = (5x)5 find f '(x)                                                                                                         3.let g(x) = 1/ (x3+x2)3       find g'(x)      4.) let y = (x3 +x2) 1/3 Find y'

Q-3. If f'(x) = -f(x) and f(1) = 1 then find f(x)

21. Suppose that functions f and g and their first derivatives have the following values at x = -1 and at x = 0. f(x) g(x)| f'(x) g'(x) 1 -1 -1 -1 -3 -2 4 Let h(x) = g(x + f(x)). Then h'(0) = (a) – 4 (b) 1 (c) – 1 (d) 0 (e) there are no such values.

Let f(x, y) = cos(x² − y), and let x = g(t) and y = h(t) be differentiable functions such that