2.07 Derivatives of Exponential and Logarithmic Functions

Find the derivative of... d/dx log 5 (x^2 sin x)

1. 1/ x^2sin In5 * d/dx (x^2 sin x) 2. 1/ (In5)x^2 sinx * (2x sinx + x^2 cosx) 3. x(2x sinx + x^2 cosx) / (In5)x^2 sinx 5. 2x sinx + x^2 cosx / (In5)xsinx = Answer HOW TO DO IT Step #1, Put this into the formula on how to find general logs for all logs, REMEMBER: log 5 is a and (x^2 sinx) is u Step #2, Put the (In 5) ahead of u. You get (2x sinx + x^2 cosx) but finding the derivative. All you had to do was find the derivative of x^2 which was 2x and leave the sinx because we didn't find the derivative for that one yet. It should look like this (2x sinx) now you have to find the derivative for sinx which is cosx. Leave the x^2 alone because we found its derivative in the first part. Now we have, (2x sinx + x^2 cosx) Step #3, The 1 turns into x and we multiplied the (2x sinx + x^2 cosx) so it's now in the numerator. Step #4, remove x multiplying (2x sinx + x^2 cosx) and divide it by x^2 in the denominator to get x.

Logarithm Rules

1. log a x = y is an inverse of a^y = x 2. log a 1 = 0 3. log a a = 1 4. log a (a^x) = x 5. a^log a x = x 6. loga (b^n) = n log a b 7. loga (1/b) = -loga b 8. loga x + loga y = loga (xy) 9. loga x - loga y = loga (x/y) 10. log x y = log y/ logx

Derivative of e^x?

ANSWER: e^x Put it in f(x+h)-f(x)/h 1. e^(x+h)-e^x / h 2. e^x *e^h -e^x 3. e^x(e^h-1) / h 4. e^x * e^h-1 / h 5. = e^x

Logarithmic Equations and Exponential Equations

Both of the graphs are inverses of each other ** Graphs look like this if the base is greater than 1 a > 1 ** Graphs between 0 and 1, is a half angle, exponential graphs go through (1,0) and logarithmic goes through (1,0) they look opposite as before

Logarithmic and Exponential form

Logarithmic form= log a x = y Exponential form= a^y = x { Both are equal to each other }

Something about logarithms

THEY MAKE THINGS SMALLER

Sketch graph of y=3-(.5)^x

This is an exponential graph **because the base .5 is less than 1 and in between 0 and 1 it will go the opposite direction. ** There is a negative sign so the graph will flip over ** The 3 moves the graph up by 3 units

How to find general log for all logs...

USE THIS FORMULA!!!! - Much easier d/dx log a u = 1/u In a * du/dx

Exponential Equations

y=c*a^x c=constant a=base which is the rate of change x=# of periods or changes This graph goes through (0,1)

Logarithmic Equations

y=c*loga x c=constant a=base x=argument Graph goes through (1,0)