Download 3.3 Derivatives of Logarithmic and Exponential Functions (10/21

3.3 Derivatives of Logarithmic and Exponential Functions
In this section we will be using the product rule, quotient rule, and chain rule to differentiate functions, but our functions will involve exponentials and logarithms, so we need to discuss their derivatives. The proofs of these can be found in your book.
First, let's review the rules from chapter 2.
Product Rule:
Ex: Find y' if Quotient Rule:
Ex: Find y' if Chain Rule:
Ex: Find y' if 1
The function f(x) = logax is differentiable and Ex: Differentiate y = log7x
Ex: Differentiate y = x3log10x
Note that by the chain rule, if our function involves loga u, where u = f (x), Ex: Differentiate y = log4 (x2 ­ 6x)
Since the natural logarithm is a specific case of the logarithm function, it is also a differentiable function, and since ln e = 1, Ex: Differentiate Ex: Differentiate 2
Note that by the chain rule, if our function involves ln (u), where u = f (x), Ex: Differentiate y = ln (x3 + 5x)
Ex: Differentiate y = ln (sin2x)
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Find the derivative of each function. f(x) = x ln x ­ x
f(x) = ln (4x2 + 5x)3
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The function f(x) = ax, a > 0, is differentiable and Ex: Differentiate y = 5x
Ex: Differentiate y = x3 + 1.4x
Since the natural exponential function is a specific case of the exponential function, it is also a differentiable function, and since ln e = 1, Ex: Differentiate y = 5x2ex
Ex: Differentiate each of the following functions:
y = 7ex + sin x ­ 4 cos x
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Note that by the chain rule, if our function involves eu, where u = f (x), Ex: Differentiate y = e3x + 4
Ex: Differentiate 7
Find y' and y'' for each of the following functions.
y = x2ln x
y = e5x + x100
Find the equation of the tangent line to the curve below at the point (0, 1).
y = e3x + 5x4
Find the equation of the tangent line to the curve below at the point (1, 5).
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Logarithmic differentiation involves taking the logarithm of both sides and then differentiating with respect to x.
Ex: Find y' if y = (sin x)x
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Ex: Use logarithmic differentiation to find y' if y = x3x ­ 4
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