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**11.2: Derivatives of Exponential and Logarithmic Functions**

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**Use the limit definition to find the derivative of ex**

Because Use graphing calculator

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

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**Example 1 Find f’(x) f(x) = 4ex – 8x2 + 7x - 14 f’(x) = 4ex – 16x + 7**

f(x) = x – x5 + e3 – x + ex f’(x) = 7x6 – 5x4 + 0 –x + ex = 7x6 – 5x –x + ex

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**Example 2 Find derivatives for A) f (x) = ex / 2 f ’(x) = ex / 2**

B) f (x) = 2ex +x2 f ’(x) = 2ex + 2x C) f (x) = -7xe – 2ex + e2 f ’(x) = -7exe-1 – 2ex Remember that e is a real number, so the power rule is used to find the derivative of xe. Also e2 is a constant, so its derivative is 0.

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**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 (log10x or log x) and natural logarithm (logex = ln x)

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**Review: properties of ln**

1) 2) 3) 4) 5)

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**Use the limit definition to find the derivative of ln x**

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

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

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Example 3 Find y’ for A) B)

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**More formulas The derivative of f(x) = bx is f’(x) = bx ln b**

The derivative of f(x) = logb x is f’(x) = Proofs are on page 598

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Example 4 Find g’(x) for A) B)

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**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) = When the demand is 400 blankets, the price is decreasing about 18 cents per blanket

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**Example 6 t = 30 corresponds to 2010 C(30) = 83 – 9 ln30 = 52.4**

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

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**Example 7: Find the equation of the tangent line to the graph of f = 2ex + 6x at x = 0**

Y = mx + b f’(x) = 2ex + 6 m = f’(0) = 2(1) + 6 = 8 y=f (0) = 2(1) + 6(0) = 2 2 = 8(0) + b so b = 2 The equation is y = 8x + 2

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**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 2ND, 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 ( , )

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Logarithmic Functions In this section, another type of function will be studied called the logarithmic function. There is a close connection between.

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