The Product and Quotient Rules
Differentiate a product or a quotient of functions whose derivatives are already known, without multiplying it out, and see where each rule comes from in the difference quotient.
What a learner can do afterwards
- Differentiate x² sin x and simplify the result
- Differentiate a quotient and say which factor belongs on top
- Show by counterexample that the derivative of a product is not the product of the derivatives
1 · Read
The derivative of u times v is u'v + uv'. Each factor takes one turn being differentiated while the other waits unchanged, then you add. Multiplying the derivatives instead is wrong: x times x is x squared with derivative 2x, not 1 times 1.
Take x squared times sin x. Call u = x squared and v = sin x, so u' = 2x and v' = cos x. The product rule gives 2x sin x + x squared cos x. The same pattern gives x cos x the derivative cos x - x sin x, since the derivative of cos x is -sin x. At x = pi/2, sin is 1 and cos is 0, so x sin x has gradient 1 there.
The derivative of u over v is (u'v - uv') / v squared. The order on top matters: derivative of the top times the bottom, minus the top times derivative of the bottom. Both rules come from the difference quotient: add and subtract one clever term, split into two limits, and the two pieces appear.
Two habits save marks. Keep the quotient order and the v squared below, since flipped order flips the sign. At a point you only need values: with f(2) = 3, f'(2) = 1, g(2) = 4 and g'(2) = 5, the product gradient is 1 times 4 + 3 times 5 = 19.
Differentiate each factor in turn for a product, and keep the order with the square for a quotient.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.