The derivative can be thought of as the slope of a curve at any given point. You can calculate the derivative using the definition, which is lim (h approaches 0) [f(x + h) - f(x)]/h. Note that there are formulas to calculate the derivative which is much simpler, but will use the definition here just so you see how this works.

Suppose f(x) = x^2 + 5x

f(x + h) = (x + h)^2 + 5(x + h)

lim (h approaches 0) [(x + h)^2 + 5(x + h) - (x^2 + 5x)]/h

= (x^2 + 2xh + h^2 + 5x +5h - x^2 - 5x)/h

= (2xh + h^2 +5h)/h

= 2x + h + 5

(put in zero for h to get) 2x + 5

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