2 (x+2)^y=y^x+2Longrightarrow y'(-1) 3. (tanx)^coty=y^e^(x)Longrightarrow y' ti

Question

Answer 1

Let's correct the approach and solve the problem step by step.Given the equation:

We need to find \( y'(-1) \).First, take the natural logarithm (ln) of both sides to make the equation easier to differentiate:

Using the properties of logarithms, we get:

Now, differentiate both sides with respect to

:

The derivative of a constant is zero, so:

Simplify the derivatives:

Combine like terms:

Factor out

:

Solve for

:

Now, substitute

into the equation:

Since \( \ln(1) = 0 \):

So, the derivative \( y'(-1) \) is:

For the second equation:

We need to find

.Take the natural logarithm of both sides:

Using the properties of logarithms:

Differentiate both sides with respect to

:

Using the product rule:

Simplify the derivatives:

Combine like terms:\[ \cot(y) \cdot \sec^2(x) - \ln(\tan(x)) \cdot \csc^2(y) \cdot y'

Soru