(x+2)^y=y^x+2Longrightarrow y'(-1)= ?

Question

Answer 1

To find the derivative of

with respect to

at the point

, we need to implicitly differentiate the given equation \( (x+2)^y = y^{x+2} \) with respect to

.Let's differentiate both sides of the equation with respect to

:

Using the chain rule on the left side, we get:

Now, let's differentiate the right side using the chain rule:

Now, we can equate the derivatives from both sides:

Now, we need to solve for

at

. Substitute

into the equation:

Now, isolate

:

Since

, we can divide both sides by \( y \cdot \ln(y) \):

Therefore, the derivative \( y'(-1) \) is equal to 0.

Soru