Animated Solution for Mathematics - Differentiation: If $2x^y+3y^x=20$, then $\frac{dy}{dx}$ at $(2,2)$ is equal to:

Imagine you are standing before a complex, winding path on a graph defined by the equation $2x^y+3y^x=20$. This is an implicit curve where $x$ and $y$ are locked in a dance, appearing in both the base and the exponent.

Our mission is to find the slope of the tangent line to this curve at the specific point $(2,2)$. In the language of calculus, we are hunting for the value of $\frac{dy}{dx}$ at that coordinate.

When you see a variable raised to a variable power, like $x^y$, the standard power rule $\frac{d}{dx}(x^n)=n{x}^{n-1}$ fails because the exponent is not a constant. We need a more robust tool.

We use the derivative formula for $u^v$, where $u$ and $v$ are both functions of $x$:

This formula is our secret weapon. It allows us to differentiate terms like $x^y$ and $y^x$ without the headache of taking logs of the entire equation.

Let us apply this to our first term, $2x^y$. Keeping the constant $2$ aside, we identify $u=x$ and $v=y$. The derivative becomes:

Now, we turn our attention to the second term, $3y^x$. Here, the base is $y$ and the exponent is $x$. Applying the same logic, we get:

Adding these together and setting the result to the derivative of the constant $20$ (which is $0$), we obtain the full differentiated equation:

Instead of expanding this massive expression, we substitute $x=2$ and $y=2$ immediately. This collapses the variables into numbers.

Substituting these values, we get:

This simplifies beautifully to:

Expanding this, we get $8+8y^{\prime }\ln 2+12y^{\prime }+12\ln 2=0$.

We group the terms containing $y^{\prime }$ on one side:

Solving for $y^{\prime }$, we get:

Factoring out a $4$ from the numerator and denominator, we find:

Finally, applying the power rule of logarithms, $n\ln a=\ln a^n$, we transform $3\ln 2$ into $\ln 8$ and $2\ln 2$ into $\ln 4$. Our final result is: