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Understanding Derivatives: Product Rule and Higher Derivatives

This resource explores how to find derivatives of products and derivatives of derivatives using the Product Rule. We begin by solving f(x) = 2cos(x) - 3sin(x) and g(x) examples. The Product Rule states that for differentiable functions f and g, the derivative of their product is (fg)'(x) = f(x)g'(x) + g(x)f'(x). We provide step-by-step examples for practical application, including higher derivatives. Practice problems are included for reinforcing concepts, making it a useful tool for mastering derivative techniques in calculus.

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Understanding Derivatives: Product Rule and Higher Derivatives

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  1. December 4, 2012 AIM: How do we find the derivative of products? Can we find derivatives of derivatives? Do Now: If f(x) = 2cosx – 3sinx + 4, find f’(x) Find g’(x) if g(x) = HW2.3a Pg. 126 - 128 #1 – 5 odd, 13, 63, 93, 94, 97

  2. How can we find the derivative of h(x) = 3x2(5x + 1)?

  3. The Product Rule How can we find the derivative of h(x) = 3x2(5x + 1)? We can use the Product Rule: If f and g are differentiable functions, then fg is a differentiable and (fg)’(x) = f(x)g’(x) + g(x)f’(x) “The derivative of a product is equal to the first function times the derivative of the second plus the second function times the derivative of the first”

  4. Example 1 Use the product rule to find the derivative of h(x) = 3x2(5x + 1) Step 1: Define f(x) and g(x) Step 2: Find f’(x) and g’(x) Step 3: Plug into formula: f(x)g’(x) + g(x)f’(x) and simplify

  5. Practice • Find • Find if • Find

  6. What if they’re combined with other derivatives? • Find f’(x) if f(x) = xsinx • Find g’(x) if g(x) = excosx • Find h’(x) if h(x) = sinxcosx

  7. Find the derivatives Find the derivative of:

  8. Higher Derivatives Now find the second derivative: And the third derivative:

  9. Practice • Find if f(x) = 4x3 – 2x + x-1 • Find f (3)(x) if f(x) = xex

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