Find the derivative of each of the following functions using the definition of derivative

Question

a) f(x)=ax+b (Assuming that a and b are non zero constants.)

Answer 1

lim h->0 [(f(x+h) - f(x)] / h is the definition of derivative, where [f(x+h) - f(x)]/h is called a difference quotient.

So we have for the difference quotient:

[a(x+h) + b - (ax+b)] /h

[ax+ah +b - ax -b]/h

ah/h

a

But the limit h->0 of a is just a.

Answer 2

There is the rule for taking the derivative of a power function which takes the th power of its input.

That is, the exponent comes down to become a coefficient in front of the thing, and the exponent is decreased by 1 .

The second rule, which is really a special case of this power-function rule, is that derivatives of constants are zero:

for any constant .

f'(x) = a