compute the derivatives of the following functions

Question

Using the limit definition of derivatives

f’(x) = lim h→0 f(x + h) − f(x)/h

compute the derivatives of the following functions

a) f(x) = 3x − 5

(b) g(x) = −x² + 9x

(c) h(x) = 2/x − 9

Answer 1

(b). g(x) = -x² + 9x

g'(x) = lim_h→0 [(g(x+h) - g(x)) / h] = lim_h→0 [(-(x+h)² + 9(x+h)) - (-x²+9x)) / h]

= lim_h→0 [(-x²-2xh-h² +9x+9h +x²-9x) / h] = lim_h→0 [(-2xh - h² + 9h) / h] = lim_h→0 (-2x - h + 9) = -2x + 9

Answer 2

a) f(x) = 3x − 5=3

(b) g(x) = −x² + 9x=-2x+9

(c) h(x) = 2/x − 9=-2/x^2