Derivative Math Example 2

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Example 2

medium
Find the derivative of f(x)=x3−4x2+7xf(x) = x^3 - 4x^2 + 7x and evaluate f′(2)f'(2).

Solution

  1. 1
    Apply the power rule term by term.
  2. 2
    For x3x^3: 3x23x^2. For −4x2-4x^2: −8x-8x. For 7x7x: 77.
  3. 3
    So f′(x)=3x2−8x+7f'(x) = 3x^2 - 8x + 7.
  4. 4
    Evaluate at x=2x = 2: f′(2)=3(4)−8(2)+7=12−16+7=3f'(2) = 3(4) - 8(2) + 7 = 12 - 16 + 7 = 3.

Answer

f′(x)=3x2−8x+7,f′(2)=3f'(x) = 3x^2 - 8x + 7, \quad f'(2) = 3
The derivative at a point gives the instantaneous rate of change. Here f′(2)=3f'(2) = 3 means the function is increasing at a rate of 3 units per unit of xx when x=2x = 2.

About Derivative

The instantaneous rate of change of a function at a single point, defined as the limit of the slope of secant lines.

Learn more about Derivative →

More Derivative Examples

Example 1 easy

Find the derivative of [formula].

Example 3 hard

Use the limit definition to find the derivative of [formula].

Example 4 easy

Find the derivative of [formula].

Example 5 hard

Find the derivative of [formula] and evaluate [formula].

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