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Section 3.2 The Product and Quotient Rules

Section 3.2 The Product and Quotient Rules. Goals Learn formulas for the derivatives of the product and quotient of two functions whose derivatives are known. The Product Rule. Suppose that f  ( x ) and g  ( x ) are each known; what is ( fg )   ( x ) ?

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Section 3.2 The Product and Quotient Rules

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  1. Section 3.2The Product andQuotient Rules • Goals • Learn formulas for the derivatives of the • product and • quotient of two functions whose derivatives are known

  2. The Product Rule • Suppose that f(x) and g(x) are each known; what is (fg) (x) ? • It is tempting to suppose that (fg) (x) = f(x)g(x) • But this is wrong! • Try it with f(x) = x and g(x) = x2 , for example.

  3. Product Rule (cont’d) • To see the correct formula, we first assume that u = f(x) and v = g(x) are both positive differentiable functions. • Then the product uv is an area of a rectangle, as shown on the next slide. • If x changes by an amount ∆x , then the corresponding changes in u and v are

  4. Product Rule (cont’d)

  5. Product Rule (cont’d) • The new value (u + ∆u)(v + ∆v) of the product is the area of the large rectangle in Fig. 1… • if ∆u and ∆v both happen to be positive. • The change in the area of the rectangle is ∆(uv) = (u + ∆u)(v + ∆v) – uv = u∆v + v∆u + ∆u∆v which is the sum of the three shaded areas.

  6. Product Rule (cont’d) • Dividing by ∆x gives • Finally we let ∆x  0 , leading to the calculations shown on the next slide. • Note that ∆u  0 as ∆x  0 since f is differentiable and therefore continuous.

  7. Product Rule (cont’d)

  8. Product Rule (cont’d) • We have assumed that all quantities involved above are positive, however… • …the calculations are valid even if not. • This leads to:

  9. Product Rule • In words, the Product Rule says that • The derivative of a product of two functions is • the first function times the derivative of the second • plus the second function times the derivative of the first.

  10. Example • If f(x) = xex , find • f(x) ; • the nth derivative, f(n)(x) . • Solution We use the Product Rule: • First,

  11. Solution (cont’d) • Then Applying the Product Rule further gives • f(x) = (x + 3)ex • f(4)(x) = (x + 4)ex In general, f(n)(x) = (x + n)ex

  12. Example • If where g(4) = 2 and g(4) = 3 , find f(4) . • Solution The Product Rule gives

  13. The Quotient Rule • We find a rule for differentiating the quotient of two differentiable functionsu = f(x) and v = g(x) in much the same way as for the Product Rule. • If x , u , and v change by amounts ∆x , ∆u , and ∆v , then the corresponding change in the quotient u/v is:

  14. Quotient Rule (cont’d) • So

  15. Quotient Rule (cont’d) • Once again, as ∆x  0 , ∆v  0 also, because g is differentiable and therefore continuous. • Thus the Limit Laws give • This leads to the Quotient Rule:

  16. Quotient Rule (cont’d) • So, the derivative of a quotient is • the denominator times the derivative of the numerator • minus the numerator times the derivative of the denominator, • all divided by the square of the denominator.

  17. Example • Find y if • Solution The Quotient Rule gives

  18. Example • Find an equation of the tangent line to the curve y = ex/(1 + x2) at the point (1, e/2) . • Solution According to the Quotient Rule,

  19. Solution (cont’d) • So the slope of the tangent line at (1, e/2) is • Thus the tangent line… • is horizontal, and • its equation is y = e/2 . • The curve and tangent line are graphed on the next slide:

  20. Solution (cont’d)

  21. Table of Formulas • Here is a summary of our differentiation formulas so far:

  22. Review • Two ways to find the derivative of combinations of differentiable functions: • The Product Rule • The Quotient Rule

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