Consider the function f(x)=4-x^2 at the point (3,-5) find the slope of the tangent line to the curve using the definition of the derivative

Answer 1

It will be lim h>0 [f(x+h)-f(x)]/h with x = 3

f(x) = 4 -x²

so f(3+h) = 4 - (3+h)² = 4 - (9 + 2h + h²) = -5 -2h -h²

f(3) = 4 - 9 = -5

So f(3+h)- f(3) = -2h - h²

Dividing this by h gives -2 - h

as h>0 this tends to -2-0 = -2

so the slope of the tangent line to the curve = derivative = -2