14 Questions
If $f(x) = 2x + 1$ and $g(x) = x^2 - 3$, what is the value of $(f \circ g)'(2)$?
$26$
What is the domain of the function $f(x) = \frac{1}{(x-2)(x-3)}$?
$(-\infty, 2) \cup (2, 3) \cup (3, \infty)$
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 $|(3x - 4) - 2| < \epsilon$?
$\delta = \frac{\epsilon}{3}$
Let $f(x) = x^2 \sin(\frac{1}{x})$ for $x \neq 0$ and $f(0) = 0$. Which of the following statements is true?
$f$ is continuous at $x = 0$ but not differentiable.
What is the derivative of the implicitly defined function $x^2 + y^2 = 4$ with respect to $x$?
$-\frac{x}{y}$
If $u$ and $v$ are functions of $x$, which of the following is equivalent to the derivative of $u/v$ with respect to $x$?
$$rac{v * du/dx - u * dv/dx}{v^2}$$
If $u$, $v$, and $w$ are functions of $x$, what is the derivative of $u(v+w)$ with respect to $x$?
$$u * (dv/dx + dw/dx) + (v+w) * du/dx$$
If $y = (x^2 + 1)(x^3 - 2)$, what is the derivative of $y$ with respect to $x$?
$$2x(x^3 - 2) + 3x^2(x^2 + 1)$$
If $y = f(g(x))$, where $f(u) = u^2$ and $g(x) = 2x + 1$, what is the derivative of $y$ with respect to $x$?
$$4(2x + 1)^2 * 2$$
If $y = (2x + 1)^3$, what is the derivative of $y$ with respect to $x$?
$$3(2x + 1)^2 * 2$$
If $y = x^2 * \sin(x)$, what is the derivative of $y$ with respect to $x$?
$$2x * \sin(x) + x^2 * \cos(x)$$
If $y = rac{x^2 + 1}{x - 1}$, what is the derivative of $y$ with respect to $x$?
$$rac{(x - 1)(2x) - (x^2 + 1)}{(x - 1)^2}$$
If $y = (x^2 + 1)(x^3 - 2)$, what is the derivative of $y$ with respect to $x$?
$$2x(x^3 - 2) + 3x^2(x^2 + 1)$$
If $y = x^n$, where $n$ is a constant, what is the derivative of $y$ with respect to $x$?
$$nx^{n-1}$$
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=1nu1∗...∗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=dundy∗dun−1dun∗...∗du1du2∗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)
Test your knowledge of calculus concepts, including composition of functions, domain and continuity, limits, epsilon delta definition, and differentiation rules. Practice problems cover implicit differentiation, mean value theorem, and more.
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