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Calculating the Derivative Continued

Calculating the Derivative Continued. We are going to learn how to find certain derivatives. We are not going to go through the proofs of the formulas. We are just going to accept them as being true. DERIVATIVE OF e x : D x [ e x ] = e x. Simply put e x is its own derivative.

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Calculating the Derivative Continued

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  1. Calculating the Derivative Continued We are going to learn how to find certain derivatives. We are not going to go through the proofs of the formulas. We are just going to accept them as being true.

  2. DERIVATIVE OF ex: Dx [ex] = ex Simply put ex is its own derivative. DERIVATIVE OF e g (x): Dx [eg (x)] = e g (x)· g'(x) Example: Find the derivative of Solution: Let g(x) = 3x 2– 4x + 2, then g'(x) = 6x – 4 making Always copy e exactly like it is in the problem.

  3. DERIVATIVE OF ax Note: a is some number such as a 7 Dx [ax] = ( ln a)a x This is the letter L not a 1. ln stands for natural logarithms. Example: Find the derivative of y = 3 x Solution: y' = ( ln 3 ) 3x Note: you can not multiply the threes together. DERIVATIVE OF ag (x) Dx [a g (x)] = ( ln a)a xg'(x) I prefer writing: Dx [ ag (x) ] = ( ln a)[g'(x)] ax Example: Find the derivative of Solution: Let g(x) = 2x 2+ 5x – 4, then g'(x) = 4x + 5 making

  4. base DERIVATIVE OF log a x Note: If no base is shown then it is an understood 10. The | | symbol may or may not appear in the problem. It stands for absolute value and we can only take the logarithm or natural logarithm of a positive quantity. If it is missing, we will assume the quantity to be positive. Example 1: Find the derivative of y = log 5x Example 2: Find the derivative of y = log x

  5. DERIVATIVE OF log a | g(x)| Notice even though the problem involves logarithms, the answer is in terms of natural logarithms. Example 1: Find the derivative of y = log 7 (3x 2– 4x + 5) Solution: Let g(x) = 3x 2– 4x + 5, then g'(x) = 6x – 4 making

  6. Example 2: Find the derivative of y = log 8 (5x 2– 3x + 6) 3 Solution: Let g(x) = (5x 2– 3x + 6 ) 3, then g'(x) = 3(10x – 3)(5x 2 – 3x + 6) 2 making (5x 2– 3x + 6) Note: In finding g'(x) above, I had to use the chain rule alternative form. That is going to be part of your problem, determining which different rule or rules to use and when to use them.

  7. DERIVATIVE OF ln| x| Example: Find the derivative of y = ln|x| You simply memorize the answer. DERIVATIVE OF ln| g(x)| Example: Find the derivative of y = ln|4x 2+ 9x + 2| Solution: Let g(x) = (4x 2+ 9x + 2 ), then g'(x) = 8x + 9 making

  8. In the next presentation, I will just work several different examples. I do not want these presentations to become extremely long.

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