Use the definition of derivative to find F ' (x) given that

Question

Use the definition of derivative to find F ' (x) given that F(x)=2x²-5x+3 (do NOT use the short cut method). Use the result to find the slope of the tangent line at the point (-1, 10).

Answer 1

U have to use the chain rule(IDK what it is called at other schools but it is basically like this

F(x + h) - F(x) = 2(x + h)² - 5(x + h) + 3 - [2x² - 5x + 3]

= 2(x² + 2xh + h²) - 5x - 5h + 3 - 2x² + 5x - 3

= 2x² + 4xh + 2h² - 5h - 2x²

= 4xh+ 2h² - 5h

Reply if you want me to continue more

ANSWER: -9

Answer 2

To find F'(x) using the limit definition, we first simply the ratio [F(x + h) - F(x)]/h, and then take the limit as h -> 0. Now

F(x + h) - F(x) = 2(x + h)² - 5(x + h) + 3 - [2x² - 5x + 3]

......................= 2(x² + 2xh + h²) - 5x - 5h + 3 - 2x² + 5x - 3

......................= 2x² + 4xh + 2h² - 5h - 2x²

......................= 4xh+ 2h² - 5h

Thus [F(x + h) - F(x)]/h = 4x + 2h - 5, which tends to 4x as h -> 0. Hence, F'(x) = 4x - 5.

The slope of the tangent line at (-1,10) is the derivative of F(x) at x = -1, i.e., F'(-1) = 4(-1) - 5 = -9.