Report

G. Derivatives of Transcendental Functions CALCULUS 30 C30.7 DEMONSTRATE UNDERSTANDING OF TRANSCENDENTAL FUNCTION DERIVATIVES AND THEIR APPLICATIONS. 1. Differentiating Logarithmic Functions C30.7 Demonstrate understanding of transcendental function derivatives and their applications. 1. Differentiating Logarithmic Functions Logarithmic functions were introduced in Math B30. We will review basic properties of logarithms, and then move in differentiating them. Laws of Logarithms 1. Product Law 2. Quotient Law 3. Power Law There are 2 special types of logarithms we could see 1. Common Logarithm – has base 10 (which is usually not written) → logx 2. Natural Logarithm – has base e (e =2.71828182845...) →lnx Example Write each of the following as the logarithm of a single term. You do not have to evaluate the logarithm. We now move on to the derivative of logarithmic functions. Examples Find the derivative of What if there is a function inside the logarithm? Our derivative changes slightly. Examples Find the derivative of Sometimes, depending on the functions given inside the logarithm, it may be easier to use the law of logarithms first, before trying to differentiate. Examples Find the derivative of the following by using logarithm laws before differentiating Finally, we may also be asked to use more complex methods to differentiate functions involving logarithms. Example Find the derivative of the function Assignment Ex. 7.1 (p. 303) #1-9 odds in each 2. Differentiating Exponential Functions C30.7 Demonstrate understanding of transcendental function derivatives and their applications. 2. Differentiating Exponential Functions An exponential function is something that looks like = You may think (based on the power rule) that the derivative of = is This is NOT the case. = −1 There are 3 situations that you will encounter when dealing with exponential functions. Example Examples Examples Assignment Ex. 7.2 (p. 311) #1-45 odds 3. Limits Involving Trigonometric Functions C30.7 Demonstrate understanding of transcendental function derivatives and their applications. 3. Limits Involving Trigonometric Functions In our study of limits, we did not touch trigonometric functions. This is because there are special properties when we look at limits involving trigonometric functions. They are: In any of the questions you will be asked, you will need to manipulate your expression into looking like one of these before evaluating the limit. Example Example Example Example Assignment Ex. 7.3 (p. 317) #1-28 odds 4. Derivatives of Sine and Cosine C30.7 Demonstrate understanding of transcendental function derivatives and their applications. 4. Derivatives of Sine and Cosine There are 2 situations for each function. One in which there is just an x inside the function, and another where there is some other function inside the trigonometric function. Examples Find the derivative of each of the following functions. = cos , = cos , ℎ = − ℎ =− sin Example Find the derivative of each function Assignment Ex. 7.4 (p. 324) #1-57 odds