Implicit Differentiation

analysis

Definition

Implicit Differentiation

Implicit differentiation is a method for computing the derivative $\frac{dy}{dx}$ when the relationship between the variables $x$ and $y$ is given implicitly by an equation of the form:

$$F(x,y)=0$$

rather than explicitly as $y=f(x)$. The method treats $y$ as an (unknown) function of $x$, differentiates every term of $F(x,y)$ with respect to $x$ while applying the chain rule to any term that involves $y$, and then solves the resulting equation for $\frac{dy}{dx}$.

Example

Let’s find $\frac{dy}{dx}$ for the equation:

$$x^3+y^3=6xy$$

1. Differentiate both sides with respect to $x$:

$$\frac{d}{dx}(x^3)+\frac{d}{dx}(y^3)=\frac{d}{dx}(6xy)$$

2. Apply the power rule and chain rule to each term:

$$3x^2+3y^2\frac{dy}{dx}=6y+6x\frac{dy}{dx}$$

3. Collect terms involving $\frac{dy}{dx}$ on one side and the remaining terms on the other side:

$$3y^2\frac{dy}{dx}-6x\frac{dy}{dx}=6y-3x^2$$

4. Factor out $\frac{dy}{dx}$ from the left side:

$$\frac{dy}{dx}(3y^2-6x)=6y-3x^2$$

6. Solve for $\frac{dy}{dx}$:

$$\frac{dy}{dx}=\frac{6y-3x^2}{3y^2-6x}$$

7. Simplify the expression by dividing numerator and denominator by 3:

$$\frac{dy}{dx}=\frac{2y-x^2}{y^2-2x}$$

Therefore, the derivative of $y$ with respect to $x$ is:

$$\boxed{\frac{2y-x^2}{y^2-2x}}$$