1. 11.2: Derivatives of Exponential and Logarithmic Functions

2. A constant function and e x are walking on Broadway. Then suddenly the constant function sees a differential operator approaching and runs away. So e x follows him and asks why the hurry. "Well, you see, there's this differential operator coming this way, and when we meet, he'll differentiate me and nothing will be left of me...!" "Ah," says e x, "he won't bother ME, I'm e to the x!" and he walks on. www.onlinemathlearning.com What is the derivative of the exponential function e x

3. Use the limit definition to find the derivative of e x Find BecauseUse graphing calculator

4. The Derivative of e x Therefore: The derivative of f (x) = e x is f ’(x) = e x.

5. Example 1 Find f’(x) A)f(x) = 4e x – 8x 2 + 7x - 14 f’(x) = 4e x – 16x + 7 B)f(x) = x 7 – x 5 + e 3 – x + e x f’(x) = 7x 6 – 5x 4 + 0 –1 + e x = 7x 6 – 5x 4 –1 + e x

6. Example 2 Remember that e is a number, so the power rule is used to find the derivative of x e. Also e 2  7.389 is a constant, so its derivative is 0. Find derivatives for A)f (x) = e x / 2 f ’(x) = e x / 2 B) f (x) = 2e x +x 2 f ’(x) = 2e x + 2x C) f (x) = -7x e – 2e x + e 2 f ’(x) = -7ex e-1 – 2e x

7. Review is equivalent to Domain: (0, ∞) Range: (-∞, ∞) Range: (0, ∞) Domain: (-∞, ∞) * These are inverse function. The graphs are symmetric with respect to the line y=x * There are many different bases for a logarithmic functions. Two special logarithmic functions are common logarithm (log 10 x or log x) and natural logarithm (log e x = ln x)

8. Review: properties of ln 1) 2) 3) 4) 5)

9. Use the limit definition to find the derivative of ln x Find Set s = h / x So when h approaches 0, s also approaches 0 Property 2 Multiply by 1 which is x / x Property 3 Definition of eProperty 4: ln(e)=1 Optional slide:

10. The Derivative of ln x Therefore: The derivative of f (x) = ln x is f ’(x) =

11. Example 3 Find y’ for A) B)

12. More formulas The derivative of f(x) = b x is f’(x) = b x ln b The derivative of f(x) = log b x is f’(x) = Proofs are on page 598

13. Example 4 Find g’(x) for A) B)

14. Example 5 An Internet store sells blankets. If the price-demand equation is p = 200(0.998) x, find the rate of change of price with respect to demand when the demand is 400 blankets and explain the result. p’ = 200 (.998) x ln(0.998) p’(400) = 200 (.998) 400 ln(0.998) = -0.18. When the demand is 400 blankets, the price is decreasing about 18 cents per blanket Try with G.C. to see if you get the same answer ZoomFit, then, xmax 400, 2 nd Trace dy/dx,

15. Example 6 A model for newspaper circulation is C(t) = 83 – 9 ln t where C is newspaper circulation (in millions) and t is the number of years (t=0 corresponds to 1980). Estimate the circulation and find the rate of change of circulation in 2010 and explain the result. t = 30 corresponds to 2010 C(30) = 83 – 9 ln30 = 52.4 C(t)’ = C’(30) = The circulation in 2010 is about 52.4 million and is decreasing at the rate of 0.3 million per year

16. Example 7: Find the equation of the tangent line to the graph of f = 2e x + 6x at x = 0 Y = mx + b f’(x) = 2e x + 6 m = f’(0) = 2(1) + 6 = 8 y=f (0) = 2(1) + 6(0) = 2 Y = mx + b 2 = 8(0) + b so b = 2 The equation is y = 8x + 2

17. Example 8: Use graphing calculator to find the points of intersection F(x) = (lnx) 2 and g(x) = x On your calculator, press Y= Type in the 2 functions above for Y1 and Y2 Press ZOOM, 6:ZStandard To have a better picture, go back to ZOOM, 2: Zoom In *Now, to find the point of intersection (there is only 1 in this problem), press 2 ND, TRACE then 5: intersect Play with the left and right arrow to find the linking dot, when you see it, press ENTER, ENTER again, then move it to the intersection, press ENTER. From there, you should see the point of intersection (.49486641,.49486641)