Exercise
$\frac{d}{dx}\left(x^x\right)\frac{x^6cosx}{sinx}$
Step-by-step Solution
Learn how to solve multiplication of integers problems step by step online. Find the derivative of x^x(x^6cos(x))/sin(x). Simplify the derivative by applying the properties of logarithms. 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}}. Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=x^{\left(6+x\right)} and g=\cos\left(x\right). The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if {f(x) = \sin(x)}, then {f'(x) = \cos(x)\cdot D_x(x)}.
Find the derivative of x^x(x^6cos(x))/sin(x)
Final answer to the exercise
$\frac{\left(\left(\ln\left(x\right)+\frac{6+x}{x}\right)x^{\left(6+x\right)}\cos\left(x\right)-x^{\left(6+x\right)}\sin\left(x\right)\right)\sin\left(x\right)-x^{\left(6+x\right)}\cos\left(x\right)^2}{\sin\left(x\right)^2}$