Exercise
$\frac{d}{dx}\left(\frac{\tan\left(x\right)}{x^2-\cot\left(x\right)}\right)^4$
Step-by-step Solution
Learn how to solve problems step by step online. Find the derivative of (tan(x)/(x^2-cot(x)))^4. The power rule for differentiation states that if n is a real number and f(x) = x^n, then f'(x) = nx^{n-1}. 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}}. The derivative of the tangent of a function is equal to secant squared of that function times the derivative of that function, in other words, if {f(x) = tan(x)}, then {f'(x) = sec^2(x)\cdot D_x(x)}. The derivative of the linear function is equal to 1.
Find the derivative of (tan(x)/(x^2-cot(x)))^4
Final answer to the exercise
$\left(\frac{\tan\left(x\right)}{x^2-\cot\left(x\right)}\right)^{3}\frac{4\left(\sec\left(x\right)^2\left(x^2-\cot\left(x\right)\right)+\left(-2x-\csc\left(x\right)^2\right)\tan\left(x\right)\right)}{\left(x^2-\cot\left(x\right)\right)^2}$