๐ Topic Summary
Implicit differentiation is a technique used to find the derivative of a function where $y$ is not explicitly defined in terms of $x$. Instead of having $y = f(x)$, you might have an equation like $x^2 + y^2 = 25$. To differentiate implicitly, you differentiate both sides of the equation with respect to $x$, treating $y$ as a function of $x$ and using the chain rule when differentiating terms involving $y$.
For example, differentiating $y^2$ with respect to $x$ gives $2y \frac{dy}{dx}$. Then, you solve for $\frac{dy}{dx}$ to find the derivative.
๐ง Part A: Vocabulary
Match the term with its definition:
| Term |
Definition |
| 1. Implicit Function |
A. The process of finding the derivative when y is not isolated. |
| 2. Chain Rule |
B. A function where y is not explicitly defined in terms of x. |
| 3. Derivative |
C. A rule that states $\frac{d}{dx}[f(g(x))] = f'(g(x)) * g'(x)$. |
| 4. Implicit Differentiation |
D. The instantaneous rate of change of a function. |
| 5. Explicit Function |
E. A function where y is isolated on one side of the equation. |
๐ Part B: Fill in the Blanks
Implicit differentiation is used when we cannot easily __________ $y$ in terms of $x$. The key is to differentiate __________ sides of the equation with respect to $x$, remembering to use the __________ rule when differentiating terms involving $y$. After differentiating, we __________ for $\frac{dy}{dx}$ to find the derivative.
๐ค Part C: Critical Thinking
Explain, in your own words, why the chain rule is essential when performing implicit differentiation. Provide an example to illustrate your explanation.