Revise later
SpaceTo flip
If confident
All Flashcards
Explain the chain rule in implicit differentiation.
When differentiating a term involving $y$, multiply by $\frac{dy}{dx}$ because $y$ is a function of $x$.
Why is $\frac{dx}{dx} = 1$?
Because the rate of change of $x$ with respect to itself is always 1.
What does $\frac{dy}{dx}$ represent graphically?
The slope of the tangent line to the curve at a given point.
What does $\frac{dy}{dx} = 0$ indicate on a graph?
A horizontal tangent line, indicating a local maximum or minimum (or saddle point).
What does the sign of $\frac{dy}{dx}$ tell you?
Positive: the function is increasing. Negative: the function is decreasing.
How to find $\frac{dy}{dx}$ using implicit differentiation?
1. Differentiate both sides with respect to $x$. 2. Apply chain rule to y terms. 3. Isolate $\frac{dy}{dx}$.
How to find the tangent line equation at a point?
1. Find $\frac{dy}{dx}$ at the point. 2. Use point-slope form: $y - y_1 = m(x - x_1)$.