lecture28 - Lecture 28 1 Lecture 28: (Sec. 5.5)...

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Unformatted text preview: Lecture 28 1 Lecture 28: (Sec. 5.5) Differentiation of Logarithmic Functions How do we find the derivative of f (x) = lnx? We have: NOTE: d (ln x) = dx Lecture 28 2 ex. Find each x-value at which the graph of f (x) = xlnx has a horizontal tangent line. Then find the relative extrema of f . Lecture 28 3 The Chain Rule for Logarithmic Functions Let f be a differentiable function of x. Then for f (x) > 0, d [ln(f (x))] = dx ex. d [ln(2x − x2)] = dx Lecture 28 4 Recall some basic Properties of the Natural Logarithm: 1) domain: 2) ln 1 = 3) ln e = 4) The Inverse Properties 5) ln (xy ) = x 6) ln ( ) = y 7) ln (xy ) = Lecture 28 ex. Find f (x) for each of the following: √ 1) f (x) = ln x 2) f (x) = ln(x2 + 1)3 3) f (x) = [ln(x2 + 1)]3 5 Lecture 28 6 ex. Find the equation of the tangent line to f (x) = ln(ln x) at x = e. Lecture 28 7 Simplifying before Differentiating ex. Find f (x) for f (x) = ln 4 2x − 1 (x + 1)3 Lecture 28 Logarithmic Differentiation dy x2 − 1 Find for y = √ dx x 2x + 4 8 Lecture 28 To find 1) 2) 3) 4) 9 dy by logarithmic differentiation: dx Lecture 28 ex. Find the derivative of f (x) = xx. NOTE: 10 Lecture 28 ex. Find the derivative of f (x) = bx, b > 0, b = 1. 11 Lecture 28 Additional Examples (Master it!) ex. Find the slope of the tangent line to ex + 1 f (x) = ln at x = ln3. x−1 e 12 Lecture 28 13 ex.√ Write the equation of the tangent line to y = e 2x 3 x3 + 1 at x = 0. 2−1 x Lecture 28 14 x ex. Find the derivative of f (x) = xe . ...
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lecture28 - Lecture 28 1 Lecture 28: (Sec. 5.5)...

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