Implicit Differentiation Classroom
56.1 Implicit Functions
Most functions we've encountered so far have been explicit functions, where $y$ is expressed directly in terms of $x$, like $y = x^2 + 3x - 5$. However, many relationships between variables are not easily (or at all) expressed in this form. These are called implicit functions, where $y$ is defined implicitly by an equation involving both $x$ and $y$.
Examples of implicit functions include $x^2 + y^2 = 25$ (a circle) or $xy = 1$ (a hyperbola). For these, it might be difficult or impossible to isolate $y$ on one side of the equation. Implicit differentiation provides a method to find $\frac{dy}{dx}$ without needing to rearrange the equation into an explicit form.
56.2 Differentiating Implicitly
The core idea of implicit differentiation is to differentiate both sides of an equation with respect to $x$, treating $y$ as a function of $x$. This means that whenever you differentiate a term involving $y$, you must apply the chain rule. For example, the derivative of $y^2$ with respect to $x$ is $2y \frac{dy}{dx}$. Similarly, the derivative of $\sin(y)$ is $\cos(y) \frac{dy}{dx}$.
After differentiating, you will have an equation containing $x$, $y$, and $\frac{dy}{dx}$. The final step is to algebraically solve this equation for $\frac{dy}{dx}$.
56.3 Products & Quotients (Implicit)
When an implicit equation involves products or quotients of terms containing both $x$ and $y$ (like $xy$ or $\frac{y}{x}$), you must apply the product rule or quotient rule in conjunction with implicit differentiation. Remember that for any term containing $y$, you'll still need to multiply by $\frac{dy}{dx}$ after differentiating with respect to $y$ (due to the chain rule).
- Product Rule: $\frac{d}{dx}(uv) = u'v + uv'$
- Quotient Rule: $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}$
Being meticulous with the application of these rules and the chain rule is key to solving these problems correctly.
56.4 Advanced Implicit Differentiation
Implicit differentiation can be applied to more complex functions involving trigonometric, exponential, or logarithmic terms with $y$, or to find higher-order derivatives like $\frac{d^2y}{dx^2}$.
When finding $\frac{d^2y}{dx^2}$, you differentiate $\frac{dy}{dx}$ implicitly with respect to $x$. This will often involve product or quotient rules again, and you'll substitute the expression for $\frac{dy}{dx}$ from the first derivative into the second derivative. This often leads to expressions for $\frac{d^2y}{dx}$ that are in terms of both $x$ and $y$.