We now give a rigorous definition of the derivative, along the lines of the definition of tangent line given above as a limit of certain secant lines.

A secant line for the function *f* (*x*) at *x* = *x*_{0} is a line through the points
(*x*_{0}, *f* (*x*_{0})) and (*x*, *f* (*x*)), for some *x* in the domain of *f*. The slope of such
a secant line is

The derivative of *f* at *x*_{0} is the
limit of the slopes of the secant lines
at *x*_{0} as *x* approaches *x*_{0} (that is, as the secant lines approach the tangent
line). Thus we have the following formula for the derivative of *f* at *x*_{0}:

f'(x_{0}) = (x_{0}) = |

If we let *Δx* = *x* - *x*_{0}, the change in *x*, then *x* = *x*_{0} + *Δx* and substitution
yields an alternate formula for the derivative:

f’(x_{0})) = (x_{0}) = |

The quotients in the above expressions are often referred to as difference quotients.

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