Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

sin(xy)=y^(2)+cos(y)

sin⁡(xy)=y2+cos⁡(y) \sin (x y)=y^{2}+\cos (y)

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Calculusright chevron icon
Find derivatives of radical functionsright chevron icon

Full solution

Q. sin⁡(xy)=y2+cos⁡(y) \sin (x y)=y^{2}+\cos (y)
  1. Identify function: Identify the function to differentiate and apply implicit differentiation. ddx[sin⁡(xy)]=ddx[y2]+ddx[cos⁡(y)]\frac{d}{dx}[\sin(xy)] = \frac{d}{dx}[y^{2}] + \frac{d}{dx}[\cos(y)]
  2. Apply implicit differentiation: Differentiate sin⁡(xy)\sin(xy) using the chain rule and product rule.\newlineddx[sin⁡(xy)]=cos⁡(xy)⋅(y+x⋅dydx)\frac{d}{dx}[\sin(xy)] = \cos(xy) \cdot (y + x \cdot \frac{dy}{dx})
  3. Differentiate sin⁡(xy)\sin(xy): Differentiate y2y^2 with respect to xx, treating yy as a function of xx.ddx[y2]=2y⋅dydx\frac{d}{dx}[y^2] = 2y \cdot \frac{dy}{dx}
  4. Differentiate y2y^2: Differentiate cos⁡(y)\cos(y) with respect to xx, treating yy as a function of xx.\newlineddx[cos⁡(y)]=−sin⁡(y)⋅dydx\frac{d}{dx}[\cos(y)] = -\sin(y) \cdot \frac{dy}{dx}
  5. Differentiate cos⁡(y)\cos(y): Combine the derivatives to form an equation.\newlinecos⁡(xy)⋅(y+xdydx)=2ydydx−sin⁡(y)dydx\cos(xy) \cdot (y + x \frac{dy}{dx}) = 2y \frac{dy}{dx} - \sin(y) \frac{dy}{dx}
  6. Combine derivatives: Solve for dydx\frac{dy}{dx}.dydx⋅(cos⁡(xy)⋅x−2y+sin⁡(y))=−cos⁡(xy)⋅y\frac{dy}{dx} \cdot (\cos(xy) \cdot x - 2y + \sin(y)) = -\cos(xy) \cdot ydydx=−cos⁡(xy)⋅ycos⁡(xy)⋅x−2y+sin⁡(y)\frac{dy}{dx} = \frac{-\cos(xy) \cdot y}{\cos(xy) \cdot x - 2y + \sin(y)}

More problems from Find derivatives of radical functions

Question
Find the derivative of f(x) f(x) .\newlinef(x)=cos⁡(x) f(x) = \cos(x) \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 1 month ago

Question
Find the derivative of f(x) f(x) .\newlinef(x)=tan⁡(x) f(x) = \tan(x) \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 2 months ago

Question
Find the derivative of f(x) f(x) .\newlinef(x)=ex f(x) = e^x \newlinef′(x)= f'\left(x\right) = ______
Get tutor helpright-arrow

Posted 2 months ago

Question
Find the derivative of f(x) f(x) .\newlinef(x)=ln⁡(x) f(x) = \ln(x) \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 2 months ago

Question
Find the derivative of f(x) f(x) .\newlinef(x)=tan⁡−1x f(x) = \tan^{-1}{x} \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 2 months ago

Question
Find the derivative of f(x) f(x) .\newlinef(x)=xex f(x) = x e^x \newlinef′(x)= f'\left(x\right) = ______
Get tutor helpright-arrow

Posted 2 months ago

Question
Find the derivative of f(x) f(x) . \newlinef(x)=exx f(x) = \frac{e^x}{x} \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 2 months ago

Question
Find the derivative of f(x) f(x) . \newline f(x)=e(x+1) f(x) = e^{(x + 1)} \newline f′(x)= f'\left(x\right) = ______
Get tutor helpright-arrow

Posted 1 month ago

Question
Find the derivative of f(x) f(x) . \newlinef(x)=x+3 f(x) = \sqrt{x+3} \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 2 months ago

Question
Find the derivative of g(x) g(x). \newline g(x)=ln⁡(2x) g(x) = \ln(2x) \newline g′(x)= g^{\prime}(x) = ______
Get tutor helpright-arrow

Posted 3 months ago

View all (84)