11 December 2021 Differentiation The product rule LO

11 December 2021 Differentiation, The product rule LO: Find the derivative of the product of functions. www. mathssupport. org

The product rule To find the derivative of a product of functions you use the product rule. The derivative of a product of functions is not equal the product of the derivatives of the functions. For example: f(x) = x 2 g(x) = x h(x) = x f ’(x) = 2 x g’(x) = 1 h’(x) = 1 www. mathssupport. org f(x) = g(x) h(x) f ’(x) ≠ g’(x) h’(x) 2 x ≠ (1)

The product rule If f(x) = u(x) v(x) Where u(x) and v(x) Are differentiable functions Then: f '(x) = u(x) v’(x) + v(x) u’(x) Another way of writing this is: If y=uv Where u and v are functions of x and differentiable Then: www. mathssupport. org

The product rule Example 1: Find f '(x) if f (x) = (2 x + 3)(4 – 3 x) Let u(x) = 2 x + 3 and v(x) = Define u and v So u’(x) = 2 Find u’(x) and v’(x) = -3 f '(x) = u(x) v’(x) + v(x) u’(x) Use the product rule f '(x) = -3 (2 x + 3) + 2 Expand brackets f '(x) = – 6 x – 9 + 8 – 6 x Simplify f '(x) = – 12 x www. mathssupport. org

The product rule Example 2: Let u = ex and v = Define u and v So Use the product rule − 2) + ex ex (8�� Removing the common factor − 2) + (4 x 2 – 2 x) ex (8�� Simplify − 2) ex (4 x 2 + 6�� www. mathssupport. org

Summary The product rule If f(x) = u(x) v(x) f '(x) = u(x) v'(x) + v(x) www. mathssupport. org If y=uv

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