Animated Solution for Mathematics - Differentiation: If $y$ is a function of $x$ and $\log (x+y)-2xy=0$, then the value of $y^{\prime }(0)$ is equal to

The path is defined by the implicit equation:

Our objective is to determine the slope of the tangent line, denoted as $y^{\prime }$, at the specific location where $x=0$.

Before calculating the slope, we must identify the corresponding $y$-coordinate at $x=0$. Substituting $x=0$ into the original equation yields:

This simplifies to $\log (y)=0$. Applying the definition of the natural logarithm, we find:

Thus, our point of interest is $(0,1)$.

To find the slope, we differentiate the entire equation with respect to $x$ using the chain rule and the product rule. Applying the operator $\frac{d}{dx}$ to both sides:

Applying the chain rule to the first term and the product rule to the second term, we obtain:

This expression represents the general relationship between the coordinates and the slope $y^{\prime }$ at any point on the curve.

Rather than isolating $y^{\prime }$ algebraically, we substitute the known coordinates $(x,y)=(0,1)$ directly into the differentiated equation:

Simplifying the arithmetic, we get:

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

We have determined that at the point $(0,1)$, the slope of the tangent line is $1$.

This indicates that the tangent line makes a perfect 45-degree angle with the $x$-axis. By breaking the problem into finding the point, applying differentiation rules, and substituting early, we successfully navigated the implicit curve.