Question

For the following exercises, use the definition for the derivative at a pointto find the derivative of the functions.

f(x) = −x2 + 4x + 7

Answer 1

The derivative of function f at x = a is given by the formula:

f\'(a) = limx→a[{f(x) – f(a)}/(x – a)]

 

Consider the Following function:

f(x) = -x2 + 4x + 7

 

Determine the derivative of function f(x) = -x2 + 4x + 7 as follows:

f\'(a) = limx→a[f(x) – f(a)/(x – 1)]

        = limx→a[(-x2 + 4x + 7) – (-a2 + 4a + 7)/(x – a)]

       = limx→a[-(x2 – a2) + 4(x – 1)/(x – a)]

       = limx→a[-(x – a)(x + a) + 4(x – a)/(x – a)]

 

Further simplify:

f\'(a) = limx→a[-(x + a) + 4]

       = -(a + a) + 4

       = -2a + 4