The Product and Quotient Rules
Apply the product and quotient rules to differentiable functions, keeping both product contributions and the quotient's subtraction order. Use the quotient rule only where its denominator is nonzero.
What a learner can do afterwards
- Apply the product rule to formulas or point data, including solving for a missing rate
- Apply the quotient rule, check its nonzero-denominator condition and simplify valid results
- Use counterexamples to test claims about products and their derivatives
1 · Read
A derivative is a limit of difference quotients. Here we use derivative rules, without epsilon-delta proofs. If u and v are differentiable at the input, (uv)' = u'v+uv'. Differentiate each factor in turn while retaining the other, then add. With angles in radians, (sin x)' = cos x and (cos x)' = -sin x. For x⁴sin x, the derivative is 4x³sin x+x⁴cos x.
To find a product derivative at a point, form its derivative before evaluating. For (2x+1)(x+4), the derivative is 2(x+4)+(2x+1). At 0 this equals 8+1 = 9. For trigonometric products, substitute the angle only after applying the product rule.
Multiplying derivatives is not the product rule. For x² times x, the product is x³ and its derivative is 3x². Multiplying the separate derivatives gives 2x, a different function. A factor's value being zero at one point also does not force the product derivative to be zero there: u'v may remain nonzero.
If u and v are differentiable at the input and v is nonzero there, (u/v)' = (u'v-uv')/v². The numerator starts with the derivative of the numerator times the denominator. Subtract the numerator times the denominator's derivative. Naming those four parts avoids reversing the subtraction.
For a complete quotient derivative, use both numerator products and square the original denominator. For cos x/(x+2), the result is [-(x+2)sin x-cos x]/(x+2)², valid for x ≠ -2. The derivatives of numerator and denominator alone cannot be divided to obtain the quotient derivative.
For h(x) = (3x+2)/(x+4), the numerator of h' is 3(x+4)-(3x+2) = 10. Thus h' = 10/(x+4)² for x ≠ -4. At 1, h' = 10/25 = 0.4. Check that the denominator is nonzero at the requested input before evaluating.
Point data suffice when the functions are differentiable there. With u(0) = 2, u'(0) = 3, v(0) = 5 and v'(0) = 4, the product derivative is 3(5)+2(4) = 23. The equation can also be solved for a missing derivative. For a quotient, use the same point data in (u'v-uv')/v² and require v ≠ 0.
A cosine product needs the negative cosine derivative. For x²cos x, the two contributions are 2x cos x and -x²sin x. Retain the original argument in each trigonometric factor. A quotient that simplifies algebraically may be differentiated in its simplified form only on the original domain.
Products add two derivative contributions. Quotients subtract in order and divide by the squared nonzero denominator.
Original product-rule and quotient-rule lesson, questions and flashcards written for this learning dataset.
Practice questions written for this learning dataset. The rewritten reading was written for this dataset. GeoGebra references retain catalog titles only. Their catalog records carry a non-commercial flag; exact source licenses remain unverified.
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Where it sits
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