Differentiation Technique #1: Logarithmic Differentiation

1. We should recall the chain rule applied to \ln\bigl(f(x)\bigr) gives us
(1)\displaystyle\frac{\mathrm{d}\ln\bigl(f(x)\bigr)}{\mathrm{d}x} = \frac{f'(x)}{f(x)}
where f'(x) = \mathrm{d}f/\mathrm{d}x. So what? Well, if f(x) is some nightmarish function (e.g. x^{x}), then we have
(2)\displaystyle f(x)\frac{\mathrm{d}\ln\bigl(f(x)\bigr)}{\mathrm{d}x} = f'(x).
Quite a quick way to compute nightmarish derivatives!

2. Example. What is the derivative of f(x)=x^{x}?

Solution: we should note that
(3)\displaystyle \ln\bigl(f(x)\bigr) = x\ln(x).
This has quite an easy derivative:
(4)\displaystyle\frac{\mathrm{d}\ln\bigl(f(x)\bigr)}{\mathrm{d}x} = \ln(x)+1
Thus we find
(5)\displaystyle f'(x)=f(x)\frac{\mathrm{d}\ln\bigl(f(x)\bigr)}{\mathrm{d}x} = x^{x}\bigl(\ln(x)+1\bigr).

3. Example. What is the derivative of g(x) = x^{\exp(x)}?

Solution: We see
(6)\displaystyle \ln\bigl(g(x)\bigr) = \exp(x)\ln(x).
This has quite an easy derivative:
(7)\displaystyle\frac{\mathrm{d}\ln\bigl(g(x)\bigr)}{\mathrm{d}x} = \exp(x)\ln(x)+\frac{\exp(x)}{x}
Thus we find
(8)\displaystyle g'(x)=g(x)\frac{\mathrm{d}\ln\bigl(g(x)\bigr)}{\mathrm{d}x} = x^{\exp(x)}\left(\exp(x)\ln(x)+\frac{\exp(x)}{x}\right).
This concludes our second example.

4. Example. What is the derviative of h(x) = (3x^{2}+5)^{1/x}?

Solution: We take its logarithm, which is
(9)\displaystyle \ln\bigl(h(x)\bigr) = \frac{1}{x}\ln\bigl(3x^{2}+5\bigr).
Now we take the derivative of the logarithm:
(10)\displaystyle\frac{\mathrm{d}\ln\bigl(h(x)\bigr)}{\mathrm{d}x} = \frac{-1}{x^{2}}\ln\bigl(3x^{2}+5\bigr)+\frac{1}{x}\frac{6x}{3x^{2}+5}.
Then we multiply both sides by h(x) to get the derivative
(11)\displaystyle h'(x)=h(x)\frac{\mathrm{d}\ln\bigl(h(x)\bigr)}{\mathrm{d}x} = \left(\frac{-1}{x^{2}}\ln\bigl(3x^{2}+5\bigr)+\frac{1}{x}\frac{6x}{3x^{2}+5}\right)(3x^{2}+5)^{1/x}.

Exercise 1. Differentiate \displaystyle y = \bigl(\sin(x)\bigr)^{x^3}.

Exercise 2. Differentiate \displaystyle y = 7x \bigl(\cos(x)\bigr)^{x/2}.

Exercise 3. Differentiate \displaystyle y = \sqrt{x}^{\sqrt{x}} \exp(x^{2}).

Exercise 4. Differentiate \displaystyle y = x^{\ln(x)} \bigl(\sec(x)\bigr)^{3x}.

Exercise 5. Differentiate \displaystyle y = \displaystyle \frac{\bigl( \ln(x)\bigr)^{x}}{2^{3x+1}} .

Exercise 6. Differentiate \displaystyle y = \displaystyle\frac{x^{2x} (x-1)^{3}}{(3+5x)^{4}}.

Exercise 7. Differentiate the function \displaystyle f(x) = \displaystyle\frac{x^{5} \exp(x)\cdot (4x+3)}{5^{\ln(x)}(3-x)^{2}}.

Exercise 8. Differentiate \displaystyle y = \displaystyle{x^{(x^{(x^4)})}}.

Acknowledgement: these examples and exercises were borrowed and/or modified from Dr Kouba’s Calculus Page, although my solutions were computed by myself.

Advertisements

About Alex Nelson

I like math. I like programming. Most of all, I love puzzles.
This entry was posted in Calculus, Differential, Natural Logarithm and tagged . Bookmark the permalink.

2 Responses to Differentiation Technique #1: Logarithmic Differentiation

  1. Pingback: [Index] Calculus of a Single Variable | My Math Blog

  2. Pingback: [Index] Calculus in a Single Variable | My Math Blog

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s