help me to ansers this questionn please

Question

Use the limit definition of the derivative to find the slope of the tangent line to the curve f(x)=5x2+2x+4 at x=2

Answer 1

the limit definition of a derivative is:

f(x + h) = 5(x + h)² + 2(x + h) + 4 = 5(x² + 2xh + h²) + 2x + 2h + 4 = 5x² + 10xh + 5h² + 2x + 2h + 4

So:

f(x + h) - f(x) = 5x² + 10xh + 5h² + 2x + 2h + 4 - (5x² + 2x + 4)

f(x + h) - f(x) = 5x² + 10xh + 5h² + 2x + 2h + 4 - 5x² - 2x - 4

f(x + h) - f(x) = 10xh + 5h² + 2h = h(10x + 5h + 2)

And (f(x + h) - f(x))/h = h(10x + 5h + 2)/h = 10x + 5h + 2

So the limit of this as h approaches zero is 10x + 2

So f '(x) = 10x + 2

meaning f '(2) = 10(2) + 2 = 22