Find the derivative of $$ \ln \left(\frac{(7 x+12)^{2}}{x^{3}+11}\right) $$.

Question

UpStudy AI · Accepted Answer

We begin with the function

$$
f(x) = \ln \left(\frac{(7x+12)^2}{x^3+11}\right).
$$

**Step 1: Use logarithm properties.**

Recall that $$\ln \left(\frac{a}{b}\right) = \ln(a) - \ln(b)$$ and $$\ln(a^2) = 2\ln(a)$$. Therefore, we can rewrite $$f(x)$$ as

$$
f(x) = 2\ln(7x+12) - \ln(x^3+11).
$$

**Step 2: Differentiate $$f(x)$$.**

Differentiate term by term.

1. Differentiate $$2\ln(7x+12)$$:

Use the chain rule:
   
   - The derivative of $$\ln(7x+12)$$ is $$\frac{1}{7x+12} \cdot 7$$.
   
   Hence,
   
   $$
   \frac{d}{dx}(2\ln(7x+12)) = 2 \cdot \frac{7}{7x+12} = \frac{14}{7x+12}.
   $$

2. Differentiate $$\ln(x^3+11)$$:

Again, using the chain rule:
   
   - The derivative of $$x^3+11$$ is $$3x^2$$.
   
   Thus,
   
   $$
   \frac{d}{dx}(\ln(x^3+11)) = \frac{3x^2}{x^3+11}.
   $$

**Step 3: Combine the results.**

Since

$$
f(x) = 2\ln(7x+12) - \ln(x^3+11),
$$

its derivative is

$$
f'(x) = \frac{14}{7x+12} - \frac{3x^2}{x^3+11}.
$$

Thus, the derivative of the original function is

$$
\boxed{\frac{14}{7x+12} - \frac{3x^2}{x^3+11}}.
$$
The derivative of $$ \ln \left(\frac{(7x+12)^2}{x^3+11}\right) $$ is $$ \frac{14}{7x+12} - \frac{3x^2}{x^3+11} $$.

Find the derivative of \( \ln \left(\frac{(7 x+12)^{2}}{x^{3}+11}\right) \).

Upstudy AI Solution

Answer

Solution

Mind Expander

Related Questions

Latest Calculus Questions