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$\frac{d}{dx}\left(\ln\left(\frac{\left(3x+22\right)^2\left(-9+e^{2x}\right)}{\sqrt[4]{\left(6x^2+2\right)^{3}}}\right)\right)$
Learn how to solve problems step by step online. Find the derivative of ln(((3x+22)^2(-9+e^(2x)))/((6x^2+2)^3^1/4)). Simplifying. 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{\left(3x+22\right)^2\left(-9+e^{2x}\right)}{\sqrt[4]{\left(6x^2+2\right)^{3}}}} 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}}.