Calculus Reference
Implicit Differentiation
Implicit differentiation finds $dy/dx$ when an equation relates $x$ and $y$ without isolating $y$.
Solved Examples
Examples cover slopes on circles and curves defined by polynomial equations.
Implicit Derivative Rules
| Expression | Derivative with respect to $x$ |
|---|---|
| $y^n$ | $ny^{n-1}y'$ |
| $xy$ | $y+xy'$ |
| $F(x,y)=0$ | Differentiate both sides, treating $y$ as a function of $x$. |
Every derivative of a term containing $y$ includes the factor $y'=dy/dx$ from the chain rule.
Circle Slope and Tangent Line
At the Point $(3,4)$ on $x^2+y^2=25$
Find the slope and tangent line at $(3,4)$.
Solution Steps
- Differentiate both sides with respect to $x$.
- Solve the equation for $y'$.
- Substitute $(3,4)$ to find the slope.
- Use point-slope form for the tangent line.
$$2x+2yy'=0$$$$y'=-\frac{x}{y}$$$$m=y'(3,4)=-\frac34$$$$\boxed{y-4=-\frac34(x-3)}$$
Polynomial Curve Slope
At the Point $(1,2)$ on $x^2+xy+y^2=7$
Find the slope of the curve at $(1,2)$.
Solution Steps
- Differentiate each term, using the product rule for $xy$.
- Collect terms containing $y'$.
- Solve for $y'$.
- Substitute $(1,2)$.
$$2x+y+xy'+2yy'=0$$$$(x+2y)y'=-(2x+y)$$$$y'=-\frac{2x+y}{x+2y}$$$$\boxed{y'(1,2)=-\frac45}$$