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0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

4. Applications of Derivatives

Implicit Differentiation

Calculus

4. Applications of Derivatives

Implicit Differentiation

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Multiple Choice

Given the equation $x e^{y} = x - y$ , find $\frac{d y}{d x}$ using implicit differentiation.

\frac{1 - e^{y}}{x e^{y} + 1}

\frac{x e^{y} + 1}{1 - e^{y}}

\frac{e^{y} + 1}{1 - x e^{y}}

\frac{1 + e^{y}}{x e^{y} - 1}

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Verified step by step guidance

Step 1: Start by recognizing that the equation x e^y = x - y involves both x and y, so implicit differentiation is required. Differentiate both sides of the equation with respect to x.

Step 2: Apply the product rule to the term x e^y on the left-hand side. Recall that the product rule states d(uv)/dx = u dv/dx + v du/dx. Here, u = x and v = e^y.

Step 3: Differentiate e^y with respect to x. Since y is a function of x, use the chain rule: d(e^y)/dx = e^y * dy/dx.

Step 4: Differentiate the right-hand side of the equation x - y. The derivative of x is 1, and the derivative of -y is -dy/dx.

Step 5: Combine all differentiated terms into a single equation, isolate dy/dx, and simplify algebraically to express dy/dx in terms of x, y, and e^y.