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Questions and Answers
If $f(x) = 2x + 1$ and $g(x) = x^2 - 3$, what is the value of $(f \circ g)'(2)$?
If $f(x) = 2x + 1$ and $g(x) = x^2 - 3$, what is the value of $(f \circ g)'(2)$?
- $26$ (correct)
- $10$
- $14$
- $30$
What is the domain of the function $f(x) = \frac{1}{(x-2)(x-3)}$?
What is the domain of the function $f(x) = \frac{1}{(x-2)(x-3)}$?
- $(-\infty, 2) \cup (2, 3) \cup (3, \infty)$ (correct)
- $(-\infty, 2) \cup (3, \infty)$
- $(-\infty, 2) \cup (2, 3) \cup [3, \infty)$
- $(-\infty, 3) \cup (2, \infty)$
Use the epsilon-delta definition to prove that $\lim_{x \to 2} (3x - 4) = 2$. What is the correct choice for $\delta$ if $|x - 2| < \delta$ and $|(3x - 4) - 2| < \epsilon$?
Use the epsilon-delta definition to prove that $\lim_{x \to 2} (3x - 4) = 2$. What is the correct choice for $\delta$ if $|x - 2| < \delta$ and $|(3x - 4) - 2| < \epsilon$?
- $\delta = \frac{\epsilon}{2}$
- $\delta = \frac{\epsilon}{3}$ (correct)
- $\delta = \epsilon$
- $\delta = 3\epsilon$
Let $f(x) = x^2 \sin(\frac{1}{x})$ for $x \neq 0$ and $f(0) = 0$. Which of the following statements is true?
Let $f(x) = x^2 \sin(\frac{1}{x})$ for $x \neq 0$ and $f(0) = 0$. Which of the following statements is true?
What is the derivative of the implicitly defined function $x^2 + y^2 = 4$ with respect to $x$?
What is the derivative of the implicitly defined function $x^2 + y^2 = 4$ with respect to $x$?
If $u$ and $v$ are functions of $x$, which of the following is equivalent to the derivative of $u/v$ with respect to $x$?
If $u$ and $v$ are functions of $x$, which of the following is equivalent to the derivative of $u/v$ with respect to $x$?
If $u$, $v$, and $w$ are functions of $x$, what is the derivative of $u(v+w)$ with respect to $x$?
If $u$, $v$, and $w$ are functions of $x$, what is the derivative of $u(v+w)$ with respect to $x$?
If $y = (x^2 + 1)(x^3 - 2)$, what is the derivative of $y$ with respect to $x$?
If $y = (x^2 + 1)(x^3 - 2)$, what is the derivative of $y$ with respect to $x$?
If $y = f(g(x))$, where $f(u) = u^2$ and $g(x) = 2x + 1$, what is the derivative of $y$ with respect to $x$?
If $y = f(g(x))$, where $f(u) = u^2$ and $g(x) = 2x + 1$, what is the derivative of $y$ with respect to $x$?
If $y = (2x + 1)^3$, what is the derivative of $y$ with respect to $x$?
If $y = (2x + 1)^3$, what is the derivative of $y$ with respect to $x$?
If $y = x^2 * \sin(x)$, what is the derivative of $y$ with respect to $x$?
If $y = x^2 * \sin(x)$, what is the derivative of $y$ with respect to $x$?
If $y = rac{x^2 + 1}{x - 1}$, what is the derivative of $y$ with respect to $x$?
If $y = rac{x^2 + 1}{x - 1}$, what is the derivative of $y$ with respect to $x$?
If $y = (x^2 + 1)(x^3 - 2)$, what is the derivative of $y$ with respect to $x$?
If $y = (x^2 + 1)(x^3 - 2)$, what is the derivative of $y$ with respect to $x$?
If $y = x^n$, where $n$ is a constant, what is the derivative of $y$ with respect to $x$?
If $y = x^n$, where $n$ is a constant, what is the derivative of $y$ with respect to $x$?
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Study Notes
Differentiation Rules
Product Rule
- The product rule is used to find the derivative of a product of two functions.
- The formula is: ddx(u∗v)=u∗dvdx+v∗dudx\frac{d}{dx} (u * v) = u * \frac{dv}{dx} + v * \frac{du}{dx}dxd​(u∗v)=u∗dxdv​+v∗dxdu​
- This rule can be generalized to find the derivative of a product of n functions: ddx(u1∗u2∗...∗un)=∑i=1nu1∗...∗ui−1∗duidx∗ui+1∗...∗un\frac{d}{dx} (u_1 * u_2 *...* u_n) = \sum_{i=1}^n u_1 *...* u_{i-1} * \frac{du_i}{dx} * u_{i+1} *...* u_ndxd​(u1​∗u2​∗...∗un​)=∑i=1n​u1​∗...∗ui−1​∗dxdui​​∗ui+1​∗...∗un​
Quotient Rule
- The quotient rule is used to find the derivative of a quotient of two functions.
- The formula is: ddx(uv)=v∗dudx−u∗dvdxv2\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v * \frac{du}{dx} - u * \frac{dv}{dx}}{v^2}dxd​(vu​)=v2v∗dxdu​−u∗dxdv​​
- This rule can be derived from the product rule and the chain rule.
Sum Rule
- The sum rule is used to find the derivative of a sum of two functions.
- The formula is: ddx(u+v)=dudx+dvdx\frac{d}{dx} (u + v) = \frac{du}{dx} + \frac{dv}{dx}dxd​(u+v)=dxdu​+dxdv​
- This rule can be generalized to find the derivative of a sum of n functions: ddx(u1+u2+...+un)=du1dx+du2dx+...+dundx\frac{d}{dx} (u_1 + u_2 +...+ u_n) = \frac{du_1}{dx} + \frac{du_2}{dx} +...+ \frac{du_n}{dx}dxd​(u1​+u2​+...+un​)=dxdu1​​+dxdu2​​+...+dxdun​​
Chain Rule
- The chain rule is used to find the derivative of a composite function.
- The formula is: dydx=dydu∗dudx\frac{dy}{dx} = \frac{dy}{du} * \frac{du}{dx}dxdy​=dudy​∗dxdu​
- This rule can be generalized to find the derivative of a composite function of n functions: dydx=dydun∗dundun−1∗...∗du2du1∗du1dx\frac{dy}{dx} = \frac{dy}{du_n} * \frac{du_n}{du_{n-1}} *...* \frac{du_2}{du_1} * \frac{du_1}{dx}dxdy​=dun​dy​∗dun−1​dun​​∗...∗du1​du2​​∗dxdu1​​
Power Rule
- The power rule is used to find the derivative of a function of the form y=xny = x^ny=xn.
- The formula is: dydx=n∗xn−1\frac{dy}{dx} = n * x^{n-1}dxdy​=n∗xn−1
- This rule can be generalized to find the derivative of a function of the form y=uny = u^ny=un, where uuu is a function of xxx: dydx=n∗un−1∗dudx\frac{dy}{dx} = n * u^{n-1} * \frac{du}{dx}dxdy​=n∗un−1∗dxdu​ (chain rule and power rule combined)
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