Implicit differentiation is a technique used to differentiate equations where $y$ is not explicitly solved for in terms of $x$. This is useful when dealing with relations between variables that cannot easily be rewritten as $y = f(x)$.

Here’s a step-by-step guide on how to perform implicit differentiation:

Steps for Implicit Differentiation:

1. Differentiate both sides of the equation with respect to $x$: Treat $y$ as a function of $x$ (i.e., $y = y(x)$) and apply the chain rule to differentiate terms involving $y$. When differentiating a term with $y$, multiply by $\frac{dy}{dx}$ (often written as $y'$).

2. Apply the product rule and chain rule where necessary: If the equation involves products or compositions of functions (like $xy$), apply the product or chain rule accordingly.

3. Collect all terms involving $\frac{dy}{dx}$ on one side of the equation.

4. Solve for $\frac{dy}{dx}$ to find the derivative.

Example:

Consider the equation $x^2 + y^2 = 25$.

1. Differentiate both sides with respect to $x$: $\frac{d}{dx}(x^2 + y^2) = \frac{d}{dx}(25)$ The derivative of $x^2$ is $2x$, and using the chain rule, the derivative of $y^2$ is $2y \cdot \frac{dy}{dx}$ (since $y$ is a function of $x$). $2x + 2y \cdot \frac{dy}{dx} = 0$

2. Solve for $\frac{dy}{dx}$: $2y \cdot \frac{dy}{dx} = -2x$ $\frac{dy}{dx} = \frac{-x}{y}$

Thus, the implicit derivative is $\frac{dy}{dx} = \frac{-x}{y}$.

Would you like more details on any specific part of this process or another example?

Relative questions:

1. How is implicit differentiation different from explicit differentiation?
2. What role does the chain rule play in implicit differentiation?
3. Can implicit differentiation be used for higher-order derivatives?
4. How would you apply implicit differentiation to trigonometric equations?
5. What is the significance of solving for $\frac{dy}{dx}$ in real-world applications?

Tip:

Always remember to apply the chain rule carefully when differentiating terms involving $y$ in implicit differentiation.