Given y = e^7x, show that dy/dx = 7e^7x using the definition of a derivative.

Answer 1

f(x+h) = exp(7(x+h))= exp(7x + 7h) = exp(7x) exp(7h)

f(x+h) - f(x) = exp(7x) exp(7h) - exp(7x)

= exp(7x) [ exp(7h)-1 ]

Difference quotient is exp(7x) [ exp(7h)-1] / h

limit as h->0 is indeterminant 0/0

exp(7x) [ 7 * exp(7h)]

limit as h->0 is now exp(7x) [ 7 * exp(0)]

= 7 * exp*(7x)