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## 3.9 Derivatives of Exponential & Logarithmic Functions

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**3.9 Derivatives of Exponential & Logarithmic Functions**What you’ll learn about Derivatives of ex, ax, ln x, logax Power Rule for Arbitrary Real Powers Why? The relationship between exponential and logarithmic functions provides a powerful differentiation tool called logarithmic differentiation.**Derivatives**Copy onto derivative sheet, then memorize.**Let’s use these formulas!**Using find Using find**At what point on the graph of the function y=3t – 4 does**the tangent line have slope 12? How can we find this out? • Find dy/dt and set it equal to zero. • Solve for t. • Evaluate f(t) in the original equation. • Hint: t is a messy number, store value for accuracy in final answer.**A line with slope m=1/a passes through the origin and is**tangent to the graph of y = ln x. What is the value of m? We know • Point of tangency has coordinates (a, ln a) for some positive a • Tangent line has slope m = 1/a • Tangent line passes through the origin so equation has point (0, 0) Find slope with formula and set it equal to 1/a. Solve for a.**Homework**Page 178 Quick Review 1-10 Exercise 3-21 (3n, nЄI)**Going the long way with the chain rule! Find**a = u(x)= u’(x) =**You try**a = u = u’ =**Using the power rule in all of its Power!**Find each derivative using**What does it mean to “Find Domain”?**If f(x) = ln (x - 6), find f ’(x). State the domain of f ’. • Find f ’ • Where is f ’ defined?**Sometimes we should use logarithms to simplify before**differentiation. Find dy/dx for y = x x • Use logs to lose the exponent, THEN take the derivative implicitly.**How Fast does H1N1 Spread?**The spread of a flu in a certain school is modeled by the equation Where P(t) is the total number of students infected t days after the flu was first noticed. Many of them may already be well again at time t. • Estimate the initial number of students infected with the flu. • How fast is the flu spreading after 3 days? • When will the flu spread at its maximum rate? What is this rate?**Homework**Page 178 Exercises 24-39 (3n, nЄI), 53, 56-62 & Study for Quiz: 3.7 thru 3.9