Calculating a deriv using logarithmic differentiation

In summary, the conversation discusses using logarithmic differentiation to calculate the derivative of a function involving logarithms, exponentials, and polynomials. The correct way to expand log terms using the ln(xy) and ln(ab) formulas is explained, and the process of taking derivatives on both sides is demonstrated. Tips on how to format equations and text in LaTeX are also provided.
  • #1
danielle36
29
0
Hello,This is my first crack at using log differentiation, but I can't seem to get too far with it...

Use logarithmic differentiation to calculate the derivative for the following function:

[tex] y = \sqrt{x}e^{x^{2}} (x^{2} + 1)^{10} [/tex]

[tex] lny = \frac{1}{2}lnx * x^{2}lne * 10ln(x^{2} + 1) [/tex]

(I'm not sure what to do with the deriv of lne^{x^{2}})

[tex] lny = \frac{1}{2} * \frac{1}{x} * [deriv of lne^{x^{2}}] * 10 *\frac{2x}{x^{2} + 1} [/tex]

[tex] lny = \frac{20x}{2x(x^{2} + 10} * [deriv of lne^{x^{2}}] [/tex]
 
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  • #2
I am assuming that you use '*' for multiplication.

You can notice that, taking derivative of summation is 'easier' than those of multiplication.

Say: (u + v + w)' = u' + v' + w' looks far easier compared with (uvw)' = u'vw + uv'w + uvw', right?

And using log function, one can change multiplication, into summation. Hence, making it more comfortable when dealing with a bunch of multiplication.

danielle36 said:
Hello,This is my first crack at using log differentiation, but I can't seem to get too far with it...

Use logarithmic differentiation to calculate the derivative for the following function:

[tex] y = \sqrt{x}e^{x^{2}} (x^{2} + 1)^{10} [/tex]

[tex] lny = \frac{1}{2}lnx * x^{2}lne * 10ln(x^{2} + 1) [/tex]

Nah, you are wrong from the first line :(, you should note that:

ln(xy) = ln(x) + ln(y), it's not ln(xy) = ln(x) * ln(y) as you've written.

I'll re-do it for you. Let's begin:

[tex]y = \sqrt{x} e ^ {x ^ 2} (x ^ 2 + 1) ^ {10}[/tex]

Take logs of both sides yields:

[tex]\Rightarrow \ln( y ) = \ln \left( \sqrt{x} e ^ {x ^ 2} (x ^ 2 + 1) ^ {10} \right)[/tex]

Use the formula: ln(xy) = ln(x) + ln(y), and ln(ab) = b ln(a) to expand all the terms on the RHS:

[tex]\Rightarrow \ln( y ) = \frac{1}{2} \ln (x) + x ^ 2 \ln (e) + 10 \ln (x ^ 2 + 1)[/tex]

ln(e) = 1, so, let's simplify it a bit:

[tex]\Rightarrow \ln( y ) = \frac{1}{2} \ln (x) + x ^ 2 + 10 \ln (x ^ 2 + 1)[/tex]

If the two functions (say, f(x), and g(x)) are the same (f(x) = g(x), for all x), then so are their derivatives (i.e, f'(x) = g'(x), for all x).

So, taking the derivatives of both sides with respect to x (not just one side as you did), we have:

[tex]\Rightarrow \frac{y'_x}{y} = \left( \frac{1}{2} \ln (x) + x ^ 2 + 10 \ln (x ^ 2 + 1) \right)'_x[/tex]

Now, what you have to do is to take the derivatives of the RHS, then multiply y over, and arrive at the desired answer.

Hope that you can go from here. :)

--------------------------------

Btw, you should put a '\' in front of a function in LaTeX to make it looks un-italics.

Compare the two:

\ln x returns: [tex]\ln x[/tex]

whereas ln x returns: [tex]ln x[/tex]

The first one looks somewhat better, eh? :)

And to insert texts in LaTeX, we use the function \mbox{your text here}.

[tex]\mbox{LaTeX is wonderful :D}[/tex] is way better than [tex]LaTeX is wonderful :D[/tex]. See? :)

Hint: Click on LaTeX images to see ther codes..
 
Last edited:

Related to Calculating a deriv using logarithmic differentiation

What is logarithmic differentiation?

Logarithmic differentiation is a process used to calculate the derivative of a function that contains logarithmic terms. It involves taking the natural logarithm of both sides of an equation, using logarithm rules to simplify, and then differentiating both sides using the chain rule.

When should logarithmic differentiation be used?

Logarithmic differentiation is most useful when a function contains both exponential and logarithmic terms, or when the function is in the form of a power raised to a variable. It can also be used when the function is difficult to differentiate using traditional methods.

What is the general process for calculating a derivative using logarithmic differentiation?

The general process for logarithmic differentiation is as follows:
1. Take the natural logarithm of both sides of the equation.
2. Use logarithm rules to simplify the expression.
3. Differentiate both sides using the chain rule.
4. Solve for the derivative by bringing the base of the logarithm back to the original function.

What are some common mistakes to avoid when using logarithmic differentiation?

Some common mistakes to avoid when using logarithmic differentiation include:
- Forgetting to take the natural logarithm of both sides of the equation.
- Forgetting to use logarithm rules to simplify the expression.
- Not using the chain rule correctly.
- Forgetting to bring the base of the logarithm back to the original function when solving for the derivative.

Can logarithmic differentiation be used for all types of functions?

No, logarithmic differentiation is most effective for functions that contain logarithmic or exponential terms. It may not be useful for other types of functions and may not yield a simpler expression or easier derivative calculation.

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