Exercise
$\frac{d}{dx}\left(cos\left(xy\right)=sin\left(x+y\right)\right)$
Step-by-step Solution
Learn how to solve problems step by step online. Find the implicit derivative d/dx(cos(xy)=sin(x+y)). Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable. 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)}. The derivative of the cosine of a function is equal to minus the sine of the function times the derivative of the function, in other words, if f(x) = \cos(x), then f'(x) = -\sin(x)\cdot D_x(x). Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=x and g=y.
Find the implicit derivative d/dx(cos(xy)=sin(x+y))
Final answer to the exercise
$y^{\prime}=\frac{-y\sin\left(xy\right)-\cos\left(x+y\right)}{x\sin\left(xy\right)+\cos\left(x+y\right)}$