Logarithmic And Exponential Differentiation

Like finding the derivative of any other function, many rules are incorporated into finding the derivative of a logarithmic or exponential function

We so far have learned 3 other basic ways of differentiating.

f(x) = ln(a*b) = ln(a) + ln(a)

f(x) = ln(a/b) = ln(x1) - ln(x2)

f(x) = lnxa = a*ln(x)…..where a is a constant

if: f(x) = ln(x)
then: f '(x) = 1/x

We know this by looking at a graph of ln(x)

1/x describes how the derivative changes as x increases. The derivative is 1/x

Now that we know the derivative, lets try some problems. If you come across anything with more than just an x. simply do a u substitution and the chain rule.

if y= ln(u) where u is a function then

y' = 1/u * u' or y= u'/u

Now, lets try some practice problems.

y = ln(3x2 +2)

y= ln ( 5x2/ 2x)

y = ln(x2 + 5)2

How did you do?

y' = 6x/3x2+2

We all know the function

y = ex

if we use the rules we know this functions derivative would be

y' = xex-1

For this we used the power rule to this equation. It may seem correct but since x is a variable not a constant, a new rule must be added. This is called Logarithmic Differentiation

To find the derivative lets take the natural log of both sides. Since we are doing the same thing to both sides, it doesnt change the value of the function.

ln y = ln ex
ln y = x * ln e
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ex= e^x^^
Wow! Fantastic!
There are four easy steps to easily differentiate log equations.

First, take the log of both sides. Then use your knowledge of logs to make th equation as simplifyed as possible. Next, differeniate both sides of the equation. Finally, simplify and cross multiply to find the derivative or y'.

Now that you know how to do it, try these practice problems!

y = 5x
y = .25x

If, you got the hang of it, try these harder ones. You just use what you just learned along with the other differentiation rules.

\$y=e^{4x^{2}+2x+3}\$
\$y=8^{2x+4}\$
\$y= 6^{sin(x)}\$

page revision: 16, last edited: 08 Jun 2010 12:46