# How do you implicitly differentiate # ysinx^2 = xe^-ysin(e^y)#?

By signing up, you agree to our Terms of Service and Privacy Policy

To implicitly differentiate ( y\sin(x^2) = x e^{-y}\sin(e^y) ), follow these steps:

- Differentiate both sides of the equation with respect to ( x ).
- Use the product rule and chain rule where necessary.
- Solve for ( \frac{dy}{dx} ) after differentiation.

After differentiation, the result is:

[ \frac{d}{dx}\left(y\sin(x^2)\right) = \frac{d}{dx}\left(x e^{-y}\sin(e^y)\right) ]

[ y\cdot\frac{d}{dx}(\sin(x^2)) + \sin(x^2)\cdot\frac{dy}{dx} = e^{-y}\sin(e^y) + x\left(e^{-y}\cdot\frac{d}{dx}(\sin(e^y)) - \sin(e^y)\cdot\frac{dy}{dx}e^{-y}\right) ]

[ y\cos(x^2)2x + \sin(x^2)\cdot\frac{dy}{dx} = e^{-y}\sin(e^y) + x\left(e^{-y}\cos(e^y)\cdot\frac{dy}{dx} - \sin(e^y)\cdot\frac{dy}{dx}e^{-y}\right) ]

[ \frac{dy}{dx}\left(\sin(x^2) - x\sin(e^y)e^{-y}\right) = e^{-y}\sin(e^y) - 2xy\cos(x^2) - xe^{-y}\cos(e^y) ]

[ \frac{dy}{dx} = \frac{e^{-y}\sin(e^y) - 2xy\cos(x^2) - xe^{-y}\cos(e^y)}{\sin(x^2) - x\sin(e^y)e^{-y}} ]

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7