The derivative of a function of f at x is given by

Question

The derivative of a function of f at x is given by

f′(x)=limh→0f(x+h)−f(x)h

provided the limit exists.

 

Use the definition of the derivative to find the derivative of f(x)=7x2+3x+4.

 

Enter the fully simplified expression for f(x+h)−f(x). Do not factor. Make sure there is a space between variables.

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

f'(x) =

Answer 1

We have f(x) = 7x² + 3x + 4 and thus f(x + h) = 7(x + h)² + 3(x + h) + 4 = 7(x² + 2xh + h²) + (3x + 3h) + 4 = 7x² + 14xh + 7h² + 3x + 3h + 4. Therefore, f(x + h) - f(x) = (7x² + 14xh + 7h² + 3x + 3h + 4) - (7x² + 3x + 4) = 7x² + 14xh + 7h² + 3x + 3h + 4 - 7x² - 3x - 4 = 14xh + 7h² + 3h = h(14x + 7h + 3). Hence, (f(x + h) - f(x)) / h = (h(14x + 7h + 3)) / h = 14x + 7h + 3. Finally, f'(x) = lim (h -> 0) ((f(x + h) - f(x)) / h) = lim (h -> 0) (14x + 7h + 3) = 14x + 7(0) + 3 = 14x +3.