Math Problem Statement

Find the derivative of y = ln(sqrt((x^2 - 11)/(x^2 + 11))).

Solution

We are asked to find the derivative of the following function:

y=ln(x211x2+11)y = \ln\left(\sqrt{\frac{x^2 - 11}{x^2 + 11}}\right)

Let's break it down step by step:

Step 1: Simplify the expression inside the logarithm

We know that ln(u)=12ln(u)\ln(\sqrt{u}) = \frac{1}{2}\ln(u), where uu is a function of xx. Applying this rule, we can rewrite the expression as:

y=12ln(x211x2+11)y = \frac{1}{2} \ln\left(\frac{x^2 - 11}{x^2 + 11}\right)

Step 2: Use the logarithmic property ln(ab)=ln(a)ln(b)\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)

This allows us to separate the terms:

y=12[ln(x211)ln(x2+11)]y = \frac{1}{2} \left[\ln(x^2 - 11) - \ln(x^2 + 11)\right]

Step 3: Differentiate using the chain rule

Now, we differentiate each part with respect to xx. Let's handle each term separately:

  1. For ln(x211)\ln(x^2 - 11), the derivative is: ddxln(x211)=1x211ddx(x211)=2xx211\frac{d}{dx} \ln(x^2 - 11) = \frac{1}{x^2 - 11} \cdot \frac{d}{dx}(x^2 - 11) = \frac{2x}{x^2 - 11}

  2. For ln(x2+11)\ln(x^2 + 11), the derivative is: ddxln(x2+11)=1x2+11ddx(x2+11)=2xx2+11\frac{d}{dx} \ln(x^2 + 11) = \frac{1}{x^2 + 11} \cdot \frac{d}{dx}(x^2 + 11) = \frac{2x}{x^2 + 11}

Step 4: Combine the results

Now, putting it all together:

dydx=12(2xx2112xx2+11)\frac{dy}{dx} = \frac{1}{2} \left( \frac{2x}{x^2 - 11} - \frac{2x}{x^2 + 11} \right)

Step 5: Factor and simplify

We can factor out the common term 2x2x:

dydx=2x2(1x2111x2+11)\frac{dy}{dx} = \frac{2x}{2} \left( \frac{1}{x^2 - 11} - \frac{1}{x^2 + 11} \right)

dydx=x(1x2111x2+11)\frac{dy}{dx} = x \left( \frac{1}{x^2 - 11} - \frac{1}{x^2 + 11} \right)

Step 6: Combine the fractions

We can combine the two fractions under a common denominator:

dydx=x((x2+11)(x211)(x211)(x2+11))\frac{dy}{dx} = x \left( \frac{(x^2 + 11) - (x^2 - 11)}{(x^2 - 11)(x^2 + 11)} \right)

Simplifying the numerator:

dydx=x(x2+11x2+11(x211)(x2+11))=x(22(x211)(x2+11))\frac{dy}{dx} = x \left( \frac{x^2 + 11 - x^2 + 11}{(x^2 - 11)(x^2 + 11)} \right) = x \left( \frac{22}{(x^2 - 11)(x^2 + 11)} \right)

Final Answer:

dydx=22x(x211)(x2+11)\frac{dy}{dx} = \frac{22x}{(x^2 - 11)(x^2 + 11)}

This is the derivative of the given function.

Would you like further details or have any questions?

5 Related Questions:

  1. How would the result change if the logarithmic function was natural log squared?
  2. Can we find the second derivative of this function?
  3. What is the domain of the original function?
  4. How does this process change for logarithms with a different base (e.g., log base 10)?
  5. How does simplifying the function before differentiation help in solving the derivative?

Tip:

When differentiating complex expressions, breaking them down using logarithmic properties and simplifying before applying the chain rule can save a lot of time and reduce errors.

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Math Problem Analysis

Mathematical Concepts

Logarithmic Differentiation
Chain Rule
Quotient Rule
Simplification

Formulas

d/dx [ln(u)] = (1/u) * (du/dx)
d/dx [sqrt(u)] = (1/2sqrt(u)) * (du/dx)
ln(a/b) = ln(a) - ln(b)

Theorems

Logarithmic Differentiation
Chain Rule

Suitable Grade Level

Undergraduate Calculus or Advanced High School Calculus