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Transcendental Functions

Transcendental Functions. Nichole uribe , alyssa kasher , jasmine clarke-telfer , dustin ness. Please refer to the formula sheet throughout the slide-show . Differentiating log functions. The derivative of the logarithmic function y = ln x is given by : ( lnx )=

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Transcendental Functions

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  1. Transcendental Functions Nichole uribe, alyssakasher, jasmine clarke-telfer, dustin ness Please refer to the formula sheet throughout the slide-show.

  2. Differentiating log functions • The derivative of the logarithmic function y = lnx is given by: • (lnx)= • For some problems, we can use the logarithmic laws to simplify our log expression before differentiating it.

  3. Example • Find the derivative of • y = ln 2x

  4. Answer • We use the log law: • log ab = log a + log b • We can write our question as: • y = ln 2x = ln 2 + ln x

  5. Answer Cont.. • Now, the derivative of a constant is 0, so • So we are left with • (from our formula above) • The final answer is…

  6. We can see from the following graph that the slope of y = ln 2x (curve in black, tangent in red) is the same as the slope of y = lnx (curve in blue, tangent in pink), at the point x = 2

  7. The u • Unfortunately, we can only use the logarithm laws to help us in a limited number of logarithm differentiation question types. • Most often, we need to find the derivative of a logarithm of some function of x. For example, we may need to find the derivative of y = 2 ln ( − 1). • We need the following formula to solve such problems:

  8. Answer • We put • u = − 1 • Then the derivative of u is given by: • u' = 6x • So the final answer is…

  9. Derivatives of exponential functions • The derivative of exis quite remarkable. The expression for the derivative is the same as the expression that we started with; that is, ex! • What does this mean? It means the slope is the same as the function value (the y-value) for all points on the graph

  10. So whats the derivative of y=3409850934580349853458903458390458345838450i834792387494939423943493789e^x?

  11. TRICK QUESTION! • The answer is… • Crazy huh?

  12. Its u again? • If u is a function of x, we can obtain the derivative of an expression in the form :

  13. example

  14. Dat log function yo • We can obtain the derivative of the logarithm function with base b using:

  15. We begin by using the following log rule to simplify our question: • We can write our question as:

  16. The first term, log26, is a constant, so its derivative is 0. The derivative of the second term is as follows, using our formula: The term on the top, log2e, is a constant. If we need a decimal value, we can work it out using change of base as follows:

  17. Integration Transcendental functions

  18. Integration of logs • The general power formula that we saw in Section 1 is valid for all values of n except n = -1. =( • We take the opposite of the derivative of the logarithmic function to solve such cases: • The (absolute value) signs around the u are necessary since the log of a negative number is not defined.

  19. Example

  20. Answer K=C=Some Constant

  21. So, tough guy. What's the answer to this question? ?

  22. Yeah, yeah, yeah. Whatever, you cheated and looked at the formula sheet.

  23. One last thing.. • Whats the answer to this?

  24. Yeah, you knew it the whole time.

  25. Welp. • That was transcendental functions. Minus all the trigonometry junkawesomeness. YOU’RE FINISHED!

  26. CONGRatulations! You’re now a transcendentalist!

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