Problem : Calculate the derivative of f (x) = x^{2} at x = 1.
Substituting 1 for x_{0} in the formula for the derivative, we havef'(1) | = | ||
= | |||
= | 2 + Δx | ||
= | 2 |
Problem : Find the vertex of the parabola f (x) = x^{2} + 2x + 2 using the derivative.
At the vertex, the tangent line to the graph will be horizontal, with slope 0. Therefore, we search for an x such that f'(x) = 0. We havef'(x) | = | limΔx→0 | |
= | limΔx→0 | ||
= | limΔx→0 | ||
= | limΔx→02x + 2 + Δx | ||
= | 2(x + 1) |
Problem : Find the equation of the tangent line to the graph of f (x) = x^{3} at x = 2.
First we compute f'(2):f'(2) | = | limΔx→0 | |
= | limΔx→0 | ||
= | 12 |
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