Exercise
$\frac{d}{dx}\ln\left(\frac{3x^2}{x^2-4}\right)$
Step-by-step Solution
Learn how to solve problems step by step online. Find the derivative of ln((3x^2)/(x^2-4)). The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If f(x)=ln\:a (where a is a function of x), then \displaystyle f'(x)=\frac{a'}{a}. Divide fractions \frac{1}{\frac{3x^2}{x^2-4}} with Keep, Change, Flip: a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}. Apply the quotient rule for differentiation, which states that if f(x) and g(x) are functions and h(x) is the function defined by {\displaystyle h(x) = \frac{f(x)}{g(x)}}, where {g(x) \neq 0}, then {\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}. Multiplying fractions \frac{x^2-4}{3x^2} \times \frac{\frac{d}{dx}\left(3x^2\right)\left(x^2-4\right)-3\frac{d}{dx}\left(x^2-4\right)x^2}{\left(x^2-4\right)^2}.
Find the derivative of ln((3x^2)/(x^2-4))
Final answer to the exercise
$\frac{-8}{x\left(x^2-4\right)}$