Aim: Differentiating Natural Log Function Course: Calculus Do Now: Aim: How do we differentiate the natural logarithmic function? Power Rule

Aim: Differentiating Natural Log Function Course: Calculus Exponential Equation y = b x Inverse Exponential Function y = log b x Logarithmic Equation Inverse of Exponential Equation x = b y Exponential example y = 2 x y = log 2 x Logarithmic example Inverse of Exponential example x = 2 y log b x = y if and only if b y = x The expression log b x is read as the “log base b of x”. The function f(x) = log b x is the logarithmic function with base b. Logarithm = Exponent y = b x “x is the logarithm of y” y = log b x “y is the logarithm of x

Aim: Differentiating Natural Log Function Course: Calculus Natural Logarithmic Function f(x) = log e x = ln x, x > 0 1. ln 1 = 0 2. ln e = 1 3. ln e x = x because e 0 = 1 because e 1 = e because e x = e x The logarithmic function with base e is called the natural log function. inverse property 5. If ln x = ln y, then x = y

Aim: Differentiating Natural Log Function Course: Calculus Properties of Natural Log 1.The domain is (0,  ) and the range is (- ,  ). 2.The function is continuous, increasing, and one-to-one. 3.The graph is concave down. If a and b are positive numbers and n is rational, then the following are true ln(1) = 0 ln(ab) = ln a + ln b ln(a n ) = n ln a ln (a/b) = ln a – ln b

Aim: Differentiating Natural Log Function Course: Calculus Using Properties of Natural Logarithms 2. ln e 2 3. ln e 0 Rewrite each expression: ln e x = x because e x = e x = -1 = 2 = ln e= 2 ln e = 1 because e 1 = e ln e x = x

Aim: Differentiating Natural Log Function Course: Calculus Model Problems Use natural logarithms to evaluate log 4 30 ` Given ln 2  0.693, ln 3  1.099, and ln 7  1.946, use the properties of logs to approximate a) ln 6b) ln 7/27 ln 6 ln 7/27 = ln (2 3) = ln 2 + ln 3   = ln 7 – ln 27 = ln 7 – 3 ln 3  – 3(1.099) 

Aim: Differentiating Natural Log Function Course: Calculus Model Problems Use properties of logarithms to rewrite = ln(3x – 5) 1/2 – ln 7 = 1/2 ln(3x – 5) – ln 7

Aim: Differentiating Natural Log Function Course: Calculus Power Rule – the exception

Aim: Differentiating Natural Log Function Course: Calculus Power Rule – the exception Power Rule don’t work!

Aim: Differentiating Natural Log Function Course: Calculus Power Rule – the exception Power Rule no antiderivative for f(x) = 1/x The domain of the natural logarithmic function is the set of all positive real numbers. Definition of the Natural Logarithmic Function 2 nd Fundamental Theorem of Calculus accumulation function

Aim: Differentiating Natural Log Function Course: Calculus Definition of the Natural Log Function ln x is positive when x > 1 x x11 ln x is negative when x < 1 ln(1) = 0 The natural log function measures the area under the curve f(x) = 1/x between 1 and x.

Aim: Differentiating Natural Log Function Course: Calculus e What is the value of x? e 1 e x (e, 1)

Aim: Differentiating Natural Log Function Course: Calculus The Derivative of the Natural Log Function Chain Rule 2 nd Fundamental Theorem of Calculus

Aim: Differentiating Natural Log Function Course: Calculus Model Problems u’ = 2

Aim: Differentiating Natural Log Function Course: Calculus Model Problems u’ = 2x

Aim: Differentiating Natural Log Function Course: Calculus Rewrite Before Differentiating rewrite u’ = 1

Aim: Differentiating Natural Log Function Course: Calculus Model Problem rewrite u = x 2 + 1u = 2x 3 – 1 u’ = 2xu’ = 6x 2

Aim: Differentiating Natural Log Function Course: Calculus Logarithmic Differentiation y is always positive therefore ln y is defined take ln of both sides Log properties Differentiate Applying the laws of logs to simplify functions that include quotients, products and/or powers can simplify differentiation.

Aim: Differentiating Natural Log Function Course: Calculus Using Log Derivative Solve for y’, Substitute for y & Simplify

Aim: Differentiating Natural Log Function Course: Calculus Derivative Involving Absolute Value Find the derivative of f(x) = ln|cosx| u = cosx u’ = -sinx

Aim: Differentiating Natural Log Function Course: Calculus Model Problem u = x 2 + 2x + 3 u’ = 2x st Derivative Test f(-1) = ln[ (-1) + 3] = ln 2 Relative Extrema – (-1, ln2) Evaluate critical point Minimum 2 nd Derivative Test

Aim: Differentiating Natural Log Function Course: Calculus Do Now: Aim: How do we differentiate the natural logarithmic function?