- Use implicit differentiation to find derivatives and the equations for tangent lines
What is Implicit Differentiation?
The Main Idea
- Implicit differentiation is a method to find derivatives of functions that are not explicitly defined in terms of one variable.
- Explicit vs. Implicit Functions:
- Explicit: [latex]y = f(x)[/latex] (e.g., [latex]y = x^2 + 1[/latex])
- Implicit: Relationship between [latex]x[/latex] and [latex]y[/latex] (e.g., [latex]x^2 + y^2 = 25[/latex])
- Treat [latex]y[/latex] as a function of [latex]x[/latex] and use the chain rule when differentiating with respect to [latex]x[/latex].
- General Process:
- Differentiate both sides of the equation with respect to [latex]x[/latex].
- Group terms with [latex]\frac{dy}{dx}[/latex] on one side.
- Solve for [latex]\frac{dy}{dx}[/latex].
Find [latex]\frac{dy}{dx}[/latex] for [latex]y[/latex] defined implicitly by the equation [latex]4x^5+ \tan y=y^2+5x[/latex].
Find [latex]\frac{dy}{dx}[/latex] for the equation [latex]x^3 + y^3 = 6xy[/latex].
Finding Tangent Lines Implicitly
The Main Idea
- Application of Implicit Differentiation:
- Used to find tangent lines for curves defined by complex equations
- Process:
- Find [latex]\frac{dy}{dx}[/latex] using implicit differentiation
- Evaluate [latex]\frac{dy}{dx}[/latex] at the given point to find the slope
- Use point-slope form to find the equation of the tangent line
- Advantages:
- Can handle curves not easily expressed as explicit functions
- Simplifies calculations for certain types of curves (e.g., conics)
- Key Equations:
- Point-slope form: [latex]y - y_1 = m(x - x_1)[/latex]
- Slope-intercept form: [latex]y = mx + b[/latex]
Find the equation of the line tangent to the hyperbola [latex]x^2-y^2=16[/latex] at the point [latex](5,3)[/latex].
Find the equation of the tangent line to the curve [latex]xy + y^2 = 4[/latex] at the point [latex](1, 1)[/latex].