Calculus Implicit Differentiation Exam Questions and Solutions

📚 Quick Study Guide

• 🔍 Implicit differentiation is used when you can't easily isolate $y$ in terms of $x$.
• 🍎 Remember to apply the chain rule when differentiating terms involving $y$ with respect to $x$. You'll get a $\frac{dy}{dx}$ term.
• 📝 The power rule states that if $y = x^n$, then $\frac{dy}{dx} = nx^{n-1}$. This still applies in implicit differentiation.
• 💡 To find $\frac{dy}{dx}$, differentiate both sides of the equation with respect to $x$, and then solve for $\frac{dy}{dx}$.
• 🧪 Common derivatives to remember: $\frac{d}{dx}(\sin x) = \cos x$, $\frac{d}{dx}(\cos x) = -\sin x$, $\frac{d}{dx}(e^x) = e^x$.
• 🔢 Product Rule: $\frac{d}{dx}(uv) = u'v + uv'$
• 📈 Quotient Rule: $\frac{d}{dx}(\frac{u}{v}) = \frac{u'v - uv'}{v^2}$

Practice Quiz

1. What is the derivative of $x^2 + y^2 = 25$ with respect to $x$?
1. $2x + 2y = 0$
2. $2x + 2y\frac{dy}{dx} = 0$
3. $2x + 2y\frac{dx}{dy} = 0$
4. $2x + 2y = 25$
2. Find $\frac{dy}{dx}$ if $x^3 + y^3 = 6xy$.
1. $\frac{2y - x^2}{y^2 - 2x}$
2. $\frac{2y - x^2}{y^2 + 2x}$
3. $\frac{2y + x^2}{y^2 - 2x}$
4. $\frac{2y + x^2}{y^2 + 2x}$
3. Given $xy + y^2 = 5$, what is $\frac{dy}{dx}$?
1. $\frac{-y}{x + 2y}$
2. $\frac{y}{x + 2y}$
3. $\frac{-y}{x - 2y}$
4. $\frac{y}{x - 2y}$
4. If $\sin(x + y) = y^2$, find $\frac{dy}{dx}$.
1. $\frac{\cos(x + y)}{2y - \cos(x + y)}$
2. $\frac{\cos(x + y)}{2y + \cos(x + y)}$
3. $\frac{-\cos(x + y)}{2y - \cos(x + y)}$
4. $\frac{-\cos(x + y)}{2y + \cos(x + y)}$
5. Determine $\frac{dy}{dx}$ for $e^{xy} = x$.
1. $\frac{1 - ye^{xy}}{xe^{xy}}$
2. $\frac{1 + ye^{xy}}{xe^{xy}}$
3. $\frac{1 - xe^{xy}}{ye^{xy}}$
4. $\frac{1 + xe^{xy}}{ye^{xy}}$
6. Find $\frac{dy}{dx}$ if $\tan(x) + \tan(y) = 1$.
1. $-\frac{\sec^2(x)}{\sec^2(y)}$
2. $\frac{\sec^2(x)}{\sec^2(y)}$
3. $-\frac{\cos^2(x)}{\cos^2(y)}$
4. $\frac{\cos^2(x)}{\cos^2(y)}$
7. What is $\frac{dy}{dx}$ for $x^2y + xy^2 = x + y$?
1. $\frac{1 - 2xy - y^2}{x^2 + 2xy - 1}$
2. $\frac{1 + 2xy + y^2}{x^2 + 2xy + 1}$
3. $\frac{1 - 2xy + y^2}{x^2 - 2xy - 1}$
4. $\frac{1 + 2xy - y^2}{x^2 - 2xy + 1}$