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lecture11

# lecture11 - Section 3.2 Derivative Rules Powers Multiples...

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Unformatted text preview: Section 3.2: Derivative Rules Powers, Multiples, Sums, Differences 1. Derivative of a Constant Function If f (x) = c for a constant c then Proof: 2. Power Rule If n is any real number, then for all x where the powers xn , xn-1 are defined. 1 Example: Find the derivatives of f (x) = x, g(x) = x, h(x) = x3 , k(x) = x2+ 3. Constant Multiple Rule If u is a differentiable function of x, and c is a constant, then Examples Find d (3x2 ), dx d (-u) dx 2 4. Derivative Sum and Difference Rules If u, v are both differentiable functions with respect to x, then at every point where both u and v are differentiable. Example 4 Find y (x) where y = x3 + 3 x2 - 5x + 1 Derivative of the Natural Exponential Function From the definition of a derivative, we have that d x e dx = limh0 3 Products and Quotients 1. Product Rule If u, v are both differentiable at x, then so is their product uv and In function notation, we have d [f (x)g(x)] dx = Examples 1 (a) Find y for y = x (x2 + ex ) (b) What is the derivative of y = (x2 + 1)(x3 + 3)? 2. Quotient Rule If u, v are differentiable at x and if v(x) = 0 then the quotient u/v is differentiable and In function notation, d dx f (x) g(x) = 4 Examples (a) Find the derivative of y = t2 -1 t2 +1 (b) Find dy/dx of y = e-x (c) What is the derivative of y = Rule? (x-1)(x2 -2x) ? x4 Do you have to use the Quotient 5 ...
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lecture11 - Section 3.2 Derivative Rules Powers Multiples...

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