Use Derivatives to Find Instantaneous Rates

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For a polynomial term ax^n, the derivative is a times n times x^(n-1). Differentiate each term, then substitute the requested input. A derivative gives a local rate, not the function's original value.

Worked example

For f(x) = x^3 + 2x, f'(x) = 3x^2 + 2. At x = 2, the tangent slope is 3 x 4 + 2 = 14.

1. For f(x) = x^2, what is f'(3)? Answer

2. A model position is s(t) = 3t^2 meters, with t in seconds. What is its instantaneous velocity at t = 2, in meters per second? Answer

3. For f(x) = 5x^3, what is f'(2)? Answer

4. What is the derivative of the constant function f(x) = 17? Answer

5. At what x does f(x) = x^2 - 6x + 5 have a horizontal tangent? Answer