# How do you implicitly differentiate #22=(y)/(1-xe^y)#?

We have;

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

To implicitly differentiate (22 = \frac{y}{1-xe^y}), we differentiate both sides of the equation with respect to (x) using the chain rule and implicit differentiation techniques.

First, differentiate (22) with respect to (x) which results in (0).

Then, differentiate (\frac{y}{1-xe^y}) with respect to (x).

The derivative of (y) with respect to (x) is (\frac{dy}{dx}).

For the denominator, (1-xe^y), differentiate (1-xe^y) with respect to (x), resulting in (-e^y\frac{dy}{dx}).

Now apply the quotient rule, where (\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}):

[ 0 = \frac{-e^y\frac{dy}{dx} \cdot y - (1-xe^y) \frac{dy}{dx}}{(1-xe^y)^2} ]

Rearrange and simplify to solve for (\frac{dy}{dx}):

[ (1-xe^y)\frac{dy}{dx} = -e^y\frac{dy}{dx} \cdot y ]

[ (1-xe^y + e^y y)\frac{dy}{dx} = 0 ]

[ \frac{dy}{dx} = 0 , \text{or} , 1-xe^y + e^y y = 0 ]

Thus, the implicit differentiation of (22 = \frac{y}{1-xe^y}) with respect to (x) yields either (\frac{dy}{dx} = 0) or (1-xe^y + e^y y = 0).

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