Implicit Differentiation
Differentiate both sides of F(x,y)=0 treating y as a function of x.
Key idea
Differentiate both sides of F(x,y)=0 treating y as a function of x.
In depth
When y is defined implicitly by an equation, differentiate both sides with respect to x, treating y as a function of x (use the chain rule, giving dy/dx), then solve for dy/dx.
Worked example
x² + y² = 25: 2x + 2y y′ = 0, so y′ = −x/y. At (3, 4) the slope is −3/4.
Real-world uses
- Finding the slope of a circle or an ellipse at any point for design and optics
Common mistake
Forgetting the dy/dx factor when differentiating terms in y.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Implicit Differentiation?
Differentiate both sides of F(x,y)=0 treating y as a function of x. It is a topic at the expert level in Calculus.
Where is Implicit Differentiation used in real life?
Real-world examples: Finding the slope of a circle or an ellipse at any point for design and optics.
What should I learn before Implicit Differentiation?
Make sure you are comfortable with Derivatives of Inverse Trig, the previous topic in Calculus.
What is a common mistake with Implicit Differentiation?
Forgetting the dy/dx factor when differentiating terms in y.
How can I practise Implicit Differentiation?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Implicit Differentiation?
Move on to Logarithmic Differentiation next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.
- Second Derivative Test – f″ > 0 means concave up; f″ < 0 concave down.