Find for the following function.

Question

Find as limit as h approaches to 0 f(x+h)−f(x)/h for the following function.

f(x)=4x^2-x

Answer 1

Since f(x) = 4x² - x, we can say that:

f(x+h) = 4(x + h)² - (x + h)

f(x+h) = 4(x² + 2xh + h²) - x - h

f(x+h) = 4x² + 8xh + 4h² - x - h

So f(x + h) - f(x) = (4x² + 8xh + 4h² - x - h) - (4x² - x)

f(x + h) - f(x) = 4x² + 8xh + 4h² - x - h - 4x² + x

f(x + h) - f(x) = 8xh + 4h² - h

f(x + h) - f(x) = h(8x + 4h - 1)

So (f(x + h) - f(x))/h = h(8x + 4h - 1)/h = 8x + 4h - 1

And the limit as h approaches zero is 8x - 1