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# The Sum, Difference, Product, and Quotient Rules for AP Calculus

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By McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for these concepts can be found at: Differentiation Practice Problems for AP Calculus

If u and v are two differentiable functions, then

### Summary of Sum, Difference, Product, and Quotient Rules

#### Example 1

Find f '(x) if f(x) = x3 – 10x + 5.

Using the sum and difference rules, you can differentiate each term and obtain f '(x) = 3x2 – 10. Or using your calculator, select the d [differentiate] function and enter d(x3 – 10x + 5, x) and obtain 3x2 – 10.

#### Example 2

If y = (3x – 5)(x4 + 8x – 1), find .

Using the product rule and v = (x4 + 8x – 1).

Then = (3)(x4 + 8x – 1) + (4x3 + 8)(3x – 5) = (3x4 + 24x – 3) + (12x4 – 20x3 + 24x – 40) = 15x4 – 20x3 + 48x – 43. Or you can use your calculator and enter d((3x – 5)(x4 + 8x – 1), x ) and obtain the same result.

#### Example 4

Using your calculator, find an equation of the tangent to the curve f(x) = x2 – 3x + 2 at x = 5.

Find the slope of the tangent to the curve at x = 5 by entering d(x2 – 3x + 2, x)|x = 5.

The result is 7. Compute f(5) = 12. Thus, the point (5, 12) is on the curve of f(x). An equation of the line whose slope m = 7 and passing through the point (5, 12) is y – 12= 7(x – 5).

Practice problems for these concepts can be found at: Differentiation Practice Problems for AP Calculus

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