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Chapter 8 Notes, Calculus I with Precalculus 3e Larson/Edwards 8.1 Chalmeta Differentiation of Exponential Functions The Number e The following limits produce the same number and we call that number e. 1 x =e lim 1 + x→∞ x lim (1 + x)1/x = e x→0 e ≈ 2.71828 . . .. e is just a number Since e is a number we can use it in the exponential function f (x) = ex . Why? Because it works in many situations. The Exponential Function The exponential function with base e is denoted by f (x) = ex where x is any real number. Properties of exponents Let a and b be positive numbers with a 6= 1, b 6= 1 and let x and y be real numbers. Then: A) Exponent Laws: 1. ax ay = ax+y 2. (ax )y = axy 3. (ab)x = ax bx a x a x 4. = x b b 5. ax = ax−y ay 1 Chapter 8 Notes, Calculus I with Precalculus 3e Larson/Edwards The derivative of the exponential function y = ex If y = ex then the derivative of y is dy = ex dx If you need to use the chain rule the derivative f (u) = eu is du d u [e ] = eu dx dx Example 8.1.1. Solve for the following derivatives. 1. y = e2x 2. y = x2 e2x 3. y = 4e3t 4. y = 3 2 +2 ex ex + 1 2 Chalmeta Chapter 8 Notes, Calculus I with Precalculus 3e Larson/Edwards 5. y = x Solve this two ways. e2x−2 6. g(t) = 2et−t 2 3 Chalmeta Chapter 8 Notes, Calculus I with Precalculus 3e Larson/Edwards 8.2 Chalmeta Logarithms and Derivatives of Logarithmic Functions Let a > 0, a 6= 1 and x > 0 then y = loga (x) ⇐⇒ x = ay The function f (x) = loga (x) is called the logarithmic function with base a OR ”log base a” The natural logarithm This is the same as before but now we use base e where e is the number we found in section 8.1. Since the log base e shows up so often we call it the natural log. loge (x) = ln(x) We also use log base 10 very often so we abbreviate that as log10 (x) = log(x). Your calculator follows the same convention. Important Properties of Logarithms Let b, M , N be positive real numbers with b 6= 1, and let p be any real number. Then: 1. logb (1) = 0 i.e. b0 = 1 2. logb (b) = 1 i.e. b1 = b 3. logb (bx ) = x i.e. bx = bx 4. blogb (x) = x if x > 0 5. logb (M N ) = logb (M ) + logb (N ) 6. logb M N = logb (M ) − logb (N ) 7. logb (M p ) = p logb (M ) 8. logb (M ) = logb (N ) ⇐⇒ M = N 4 Chapter 8 Notes, Calculus I with Precalculus 3e Larson/Edwards The derivative of the logarithmic function y = ln x If y = ln x then the derivative of y is dy 1 = dx x If you need to use the chain rule the derivative f (u) = ln u is du d [ln u] = dx u Example 8.2.1. Solve for the following derivatives. 1. y = ln 5x 2. y = x2 ln(2x + 1) 3. y = 2 ln t5 4. h(x) = ln 1 2x3 5 Chalmeta Chapter 8 Notes, Calculus I with Precalculus 3e Larson/Edwards 5. f (x) = ln x+1 2x − 2 6. g(t) = ln(3t2 − 2t + 3) 6 Chalmeta

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