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Questions and Answers
According to the derivative rules, what is the derivative of a constant c with respect to x?
According to the derivative rules, what is the derivative of a constant c with respect to x?
- `x`
- `c`
- 1
- 0 (correct)
The derivative of x with respect to x is equal to 0.
The derivative of x with respect to x is equal to 0.
False (B)
Given u is a function of x, what is the correct application of the constant multiple rule for differentiation?
Given u is a function of x, what is the correct application of the constant multiple rule for differentiation?
- `d/dx (cu) = c + du/dx`
- `d/dx (cu) = c(du/dx)` (correct)
- `d/dx (cu) = du/dx`
- `d/dx (cu) = u(dc/dx)`
State the sum/difference rule for derivatives.
State the sum/difference rule for derivatives.
According to the product rule, d/dx (uv) = u(dv/dx) + v(d______/dx).
According to the product rule, d/dx (uv) = u(dv/dx) + v(d______/dx).
What is the correct formula for the quotient rule when finding the derivative of u/v with respect to x?
What is the correct formula for the quotient rule when finding the derivative of u/v with respect to x?
Given u is a function of x, what is the power rule?
Given u is a function of x, what is the power rule?
The derivative of $\sqrt{u}$ with respect to x is $\frac{\sqrt{u}}{2u} \frac{du}{dx}$.
The derivative of $\sqrt{u}$ with respect to x is $\frac{\sqrt{u}}{2u} \frac{du}{dx}$.
What is the limit of a constant c as x approaches a?
What is the limit of a constant c as x approaches a?
As $x$ approaches $a$, the limit of $x$ is:
As $x$ approaches $a$, the limit of $x$ is:
The limit of $cf(x)$ as $x$ approaches $a$ is $c \cdot$ lim ______.
The limit of $cf(x)$ as $x$ approaches $a$ is $c \cdot$ lim ______.
The limit of $[f(x) + g(x)]$ as $x$ approaches $a$ equals to $\lim_{x \to a} f(x) - \lim_{x \to a} g(x)$.
The limit of $[f(x) + g(x)]$ as $x$ approaches $a$ equals to $\lim_{x \to a} f(x) - \lim_{x \to a} g(x)$.
What does the limit $\lim_{x \to a} [f(x) \cdot g(x)]$ equal, assuming both limits exist?
What does the limit $\lim_{x \to a} [f(x) \cdot g(x)]$ equal, assuming both limits exist?
According to limit laws, what is $\lim_{x \to a} \frac{f(x)}{g(x)}$ equal to, provided that $\lim_{x \to a} g(x) \neq 0$?
According to limit laws, what is $\lim_{x \to a} \frac{f(x)}{g(x)}$ equal to, provided that $\lim_{x \to a} g(x) \neq 0$?
$\lim_{x \to a} [f(x)]^n$ cannot be expressed as $[\lim_{x \to a} f(x)]^n$.
$\lim_{x \to a} [f(x)]^n$ cannot be expressed as $[\lim_{x \to a} f(x)]^n$.
What is the value of $\lim_{x \to \infty} \frac{1}{x}$?
What is the value of $\lim_{x \to \infty} \frac{1}{x}$?
As $x$ approaches 0, $\lim _{x \to 0} \frac{1}{x}$ approaches ______.
As $x$ approaches 0, $\lim _{x \to 0} \frac{1}{x}$ approaches ______.
Match the limit expressions with their corresponding equivalent forms, where applicable, as x approaches a constant.
Match the limit expressions with their corresponding equivalent forms, where applicable, as x approaches a constant.
If n > 1, the statement $\lim_{x \to 0} \frac{1}{x^n} = \infty $ is false.
If n > 1, the statement $\lim_{x \to 0} \frac{1}{x^n} = \infty $ is false.
The limit $\lim_{x \to a} \sqrt[n]{f(x)}$ is equal to what?
The limit $\lim_{x \to a} \sqrt[n]{f(x)}$ is equal to what?
Flashcards
d(c)/dx
d(c)/dx
The derivative of a constant is zero.
d(x)/dx
d(x)/dx
The derivative of x with respect to x is one.
d(cu)/dx
d(cu)/dx
The derivative of a constant times a function.
d(u + v)/dx
d(u + v)/dx
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d(uv)/dx
d(uv)/dx
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d[u/v]/dx
d[u/v]/dx
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d(u^n)/dx
d(u^n)/dx
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d(√u)/dx
d(√u)/dx
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d[1/u^n]/dx
d[1/u^n]/dx
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lim (c) as x -> a
lim (c) as x -> a
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lim (x) as x -> a
lim (x) as x -> a
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lim [cf(x)] as x -> a
lim [cf(x)] as x -> a
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lim f(x)/g(x) as x -> a
lim f(x)/g(x) as x -> a
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lim f(x)^n as x -> a
lim f(x)^n as x -> a
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lim nth root of f(x)
lim nth root of f(x)
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lim 1/x as x -> ∞
lim 1/x as x -> ∞
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lim 1/x as x -> 0
lim 1/x as x -> 0
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lim 1/x^n as x -> 0
lim 1/x^n as x -> 0
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Study Notes
- Study notes on derivatives and limits
Derivatives
- d(c)/dx = 0, where c is a constant
- d(x)/dx = 1
- d(cu)/dx = c * d(u)/dx, where c is a constant
- d(u + v)/dx = du/dx + dv/dx (sum/diff)
- d(uv)/dx = u(dv/dx) + v(du/dx) (product rule)
- d(u/v)/dx = (v(du/dx) - u(dv/dx)) / v^2 (quotient rule)
- d(u^n)/dx = n*u^(n-1) * (du/dx) (power rule)
- d(√u)/dx = (1/(2√u)) * (du/dx)
- d(1/u^n)/dx = (-n/u^(n+1)) * (du/dx)
Limits
- lim (c) = c as x approaches a
- lim (x) = a as x approaches a
- lim (c * f(x)) = c * lim (f(x)) as x approaches a
- lim (f(x) ± g(x)) = lim (f(x)) ± lim (g(x)) as x approaches a
- lim (f(x) * g(x)) = lim (f(x)) * lim (g(x)) as x approaches a
- lim (f(x) / g(x)) = lim (f(x)) / lim (g(x)) as x approaches a, provided lim (g(x)) ≠0
- lim (f(x))^n = [lim (f(x))]^n as x approaches a
- lim (n√f(x)) = n√lim (f(x)) as x approaches a
- lim (1/x) = 0 as x approaches ∞
- lim (1/x) = ∞ as x approaches 0
- lim (1/x^n) = ∞ as x approaches 0, n > 1
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