Implicit differentiation

Definition

Implicit differentiation is a technique used to find the derivative of a function when it is not explicitly solved for one variable in terms of another. It involves differentiating both sides of an equation with respect to the independent variable and then solving for the desired derivative.

5 Must Know Facts For Your Next Test

1. Implicit differentiation requires applying the chain rule since both dependent and independent variables are differentiated.
2. When using implicit differentiation, treat all variables as functions of the independent variable, typically $x$.
3. You must solve for $\frac{dy}{dx}$ after differentiating both sides of the equation.
4. Implicit differentiation is often used when dealing with equations that define curves, such as circles and ellipses, where solving for one variable explicitly is difficult or impossible.
5. Common mistakes include forgetting to apply the chain rule or incorrectly simplifying after differentiating.

Review Questions

• How do you apply the chain rule in implicit differentiation?
• What steps are involved in solving an implicit differentiation problem?
• Why might you use implicit differentiation instead of explicit differentiation?

Related terms

Chain Rule: A fundamental rule in calculus used to differentiate compositions of functions. It states that $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$.

Explicit Function: A function expressed directly in terms of one variable. For example, $y = f(x)$.

Partial Derivative: The derivative of a multivariable function with respect to one variable while keeping the other variables constant. Notated as $\frac{\partial f}{\partial x}$.