3.9 derivatives of exponential & logarithmic functions what you’ll learn about derivatives of...
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3.9 Derivatives of Exponential & Logarithmic Functions
What you’ll learn about
•Derivatives of ex, ax, ln x, loga x
•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.
dx
duee
dx
d uu
dx
duaaa
dx
d uu ln)(
dx
du
uu
dx
d1
ln
dx
dunuu
dx
d nn 1
dx
du
auu
dx
da
ln
1log
Let’s use these formulas!
Using find
Using find
xxedx
d 62
dx
duee
dx
d uu
dx
duaaa
dx
d uu ln)( 25 xdx
d
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) =
xa adx
d sinlog
dx
du
auu
dx
da
ln
1log
You try xdx
d 1log2
dx
du
auu
dx
da
ln
1log
a =u =u’ =
Using the power rule in all of its Power!
3xdx
d 3)4cos4( xdx
d
Find each derivative using dx
dunuu
dx
d nn 1
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 fluwas first noticed. Many of them may already be well again at time t.
a) Estimate the initial number of students infected with the flu.b) How fast is the flu spreading after 3 days?c) When will the flu spread at its maximum rate? What is this rate?
tetP
31
100)(
Homework
Page 178
Exercises 24-39 (3n, nЄI),
53, 56-62
&
Study for Quiz: 3.7 thru 3.9