Rates of Change in Applied Concepts - AP Calculus AB
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What is the derivative of $h(x) = \frac{1}{x^5}$?
What is the derivative of $h(x) = \frac{1}{x^5}$?
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$-\frac{5}{x^6}$. Power rule: $x^{-5}$ becomes $-5x^{-6}$.
$-\frac{5}{x^6}$. Power rule: $x^{-5}$ becomes $-5x^{-6}$.
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Determine the derivative of the function $f(x) = 5x^4 - 3x^2 + 7$.
Determine the derivative of the function $f(x) = 5x^4 - 3x^2 + 7$.
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$20x^3 - 6x$. Power rule applied: $20x^3$ from $5x^4$ and $-6x$ from $-3x^2$.
$20x^3 - 6x$. Power rule applied: $20x^3$ from $5x^4$ and $-6x$ from $-3x^2$.
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What is the derivative of $f(x) = x^3 - 2x + 4$ with respect to $x$?
What is the derivative of $f(x) = x^3 - 2x + 4$ with respect to $x$?
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$3x^2 - 2$. Power rule: $3x^2$ from $x^3$ and $-2$ from $-2x$.
$3x^2 - 2$. Power rule: $3x^2$ from $x^3$ and $-2$ from $-2x$.
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What is the formula for the rate of change of the volume of a sphere with respect to its radius?
What is the formula for the rate of change of the volume of a sphere with respect to its radius?
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$4\pi r^2$. Derivative of sphere volume $\frac{4}{3}\pi r^3$.
$4\pi r^2$. Derivative of sphere volume $\frac{4}{3}\pi r^3$.
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What is the derivative of $g(x) = x^3 + \frac{1}{x}$?
What is the derivative of $g(x) = x^3 + \frac{1}{x}$?
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$3x^2 - \frac{1}{x^2}$. Power rule: $3x^2$ from $x^3$ and $-x^{-2}$ from $x^{-1}$.
$3x^2 - \frac{1}{x^2}$. Power rule: $3x^2$ from $x^3$ and $-x^{-2}$ from $x^{-1}$.
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What is the rate of change of the surface area of a sphere with respect to its radius?
What is the rate of change of the surface area of a sphere with respect to its radius?
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$8\pi r$. Derivative of sphere surface area $4\pi r^2$.
$8\pi r$. Derivative of sphere surface area $4\pi r^2$.
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What is the derivative of the function $f(x) = x^5 - 4x^3 + 2$?
What is the derivative of the function $f(x) = x^5 - 4x^3 + 2$?
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$5x^4 - 12x^2$. Power rule: $5x^4$ from $x^5$ and $-12x^2$ from $-4x^3$.
$5x^4 - 12x^2$. Power rule: $5x^4$ from $x^5$ and $-12x^2$ from $-4x^3$.
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What is the rate of change of the perimeter of a square with respect to its side length?
What is the rate of change of the perimeter of a square with respect to its side length?
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- Each side contributes 1 to perimeter, so total rate is 4.
- Each side contributes 1 to perimeter, so total rate is 4.
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Find the rate of change of the circumference of a circle with respect to its radius.
Find the rate of change of the circumference of a circle with respect to its radius.
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$2\pi$. Derivative of circumference $2\pi r$ is constant $2\pi$.
$2\pi$. Derivative of circumference $2\pi r$ is constant $2\pi$.
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What is the derivative of $y = x^3 + x^2$ with respect to $x$?
What is the derivative of $y = x^3 + x^2$ with respect to $x$?
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$3x^2 + 2x$. Apply power rule: $3x^2$ from $x^3$ and $2x$ from $x^2$.
$3x^2 + 2x$. Apply power rule: $3x^2$ from $x^3$ and $2x$ from $x^2$.
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Identify the derivative of $y = 3x^3 + 5x^2 - x$.
Identify the derivative of $y = 3x^3 + 5x^2 - x$.
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$9x^2 + 10x - 1$. Power rule: $9x^2$ from $3x^3$, $10x$ from $5x^2$, $-1$ from $-x$.
$9x^2 + 10x - 1$. Power rule: $9x^2$ from $3x^3$, $10x$ from $5x^2$, $-1$ from $-x$.
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What is the derivative of $h(x) = \frac{1}{x^2}$?
What is the derivative of $h(x) = \frac{1}{x^2}$?
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$-\frac{2}{x^3}$. Power rule: $x^{-2}$ becomes $-2x^{-3}$.
$-\frac{2}{x^3}$. Power rule: $x^{-2}$ becomes $-2x^{-3}$.
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What is the derivative of the function $f(x) = \frac{2}{x^3}$?
What is the derivative of the function $f(x) = \frac{2}{x^3}$?
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$-\frac{6}{x^4}$. Power rule: $2x^{-3}$ becomes $-6x^{-4}$.
$-\frac{6}{x^4}$. Power rule: $2x^{-3}$ becomes $-6x^{-4}$.
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Find the rate of change of the volume of a cylinder with respect to its height.
Find the rate of change of the volume of a cylinder with respect to its height.
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$\pi r^2$. For cylinder volume $\pi r^2 h$, derivative with respect to height.
$\pi r^2$. For cylinder volume $\pi r^2 h$, derivative with respect to height.
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Identify the rate of change of the volume of a cube with respect to its side length.
Identify the rate of change of the volume of a cube with respect to its side length.
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$3s^2$. Derivative of volume $s^3$ with respect to side length.
$3s^2$. Derivative of volume $s^3$ with respect to side length.
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What is the derivative of the function $g(x) = x^4 - \frac{1}{x^2}$?
What is the derivative of the function $g(x) = x^4 - \frac{1}{x^2}$?
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$4x^3 + \frac{2}{x^3}$. Power rule: $4x^3$ from $x^4$ and $2x^{-3}$ from $-x^{-2}$.
$4x^3 + \frac{2}{x^3}$. Power rule: $4x^3$ from $x^4$ and $2x^{-3}$ from $-x^{-2}$.
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Find the derivative of $h(x) = 7x^5 - 3x^2 + 1$.
Find the derivative of $h(x) = 7x^5 - 3x^2 + 1$.
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$35x^4 - 6x$. Power rule: $35x^4$ from $7x^5$ and $-6x$ from $-3x^2$.
$35x^4 - 6x$. Power rule: $35x^4$ from $7x^5$ and $-6x$ from $-3x^2$.
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Determine the derivative of the function $y = \frac{2}{x^2}$.
Determine the derivative of the function $y = \frac{2}{x^2}$.
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$-\frac{4}{x^3}$. Power rule: $2x^{-2}$ becomes $-4x^{-3}$.
$-\frac{4}{x^3}$. Power rule: $2x^{-2}$ becomes $-4x^{-3}$.
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Identify the rate of change of the volume of a rectangular prism with respect to its height.
Identify the rate of change of the volume of a rectangular prism with respect to its height.
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Area of base. Volume formula is base area times height.
Area of base. Volume formula is base area times height.
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Find the derivative of $y = 2x^3 + 3x^2 - 5x + 1$.
Find the derivative of $y = 2x^3 + 3x^2 - 5x + 1$.
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$6x^2 + 6x - 5$. Power rule applied: $6x^2$, $6x$, and $-5$ from each term.
$6x^2 + 6x - 5$. Power rule applied: $6x^2$, $6x$, and $-5$ from each term.
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What is the derivative of $f(x) = 2x^3 - x^2 + 3$?
What is the derivative of $f(x) = 2x^3 - x^2 + 3$?
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$6x^2 - 2x$. Power rule: $6x^2$ from $2x^3$ and $-2x$ from $-x^2$.
$6x^2 - 2x$. Power rule: $6x^2$ from $2x^3$ and $-2x$ from $-x^2$.
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Identify the rate of change of the area of a rectangle with respect to its width.
Identify the rate of change of the area of a rectangle with respect to its width.
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Length. For rectangle area $lw$, derivative with respect to width is length.
Length. For rectangle area $lw$, derivative with respect to width is length.
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What is the derivative of the function $g(x) = \frac{1}{x^4}$?
What is the derivative of the function $g(x) = \frac{1}{x^4}$?
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$-\frac{4}{x^5}$. Power rule: $x^{-4}$ becomes $-4x^{-5}$.
$-\frac{4}{x^5}$. Power rule: $x^{-4}$ becomes $-4x^{-5}$.
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Identify the derivative of the function $g(x) = \frac{1}{x}$.
Identify the derivative of the function $g(x) = \frac{1}{x}$.
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$-\frac{1}{x^2}$. Power rule: $x^{-1}$ becomes $-x^{-2}$.
$-\frac{1}{x^2}$. Power rule: $x^{-1}$ becomes $-x^{-2}$.
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Identify the derivative of $y = 6x^2 + 4x - 5$.
Identify the derivative of $y = 6x^2 + 4x - 5$.
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$12x + 4$. Power rule: $12x$ from $6x^2$ and $4$ from $4x$.
$12x + 4$. Power rule: $12x$ from $6x^2$ and $4$ from $4x$.
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What is the rate of change of the area of a rectangle with respect to its length?
What is the rate of change of the area of a rectangle with respect to its length?
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Width. For rectangle area $lw$, derivative with respect to length is width.
Width. For rectangle area $lw$, derivative with respect to length is width.
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Determine the derivative of $f(x) = 4x^2 - 3x + 7$.
Determine the derivative of $f(x) = 4x^2 - 3x + 7$.
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$8x - 3$. Power rule: $8x$ from $4x^2$ and $-3$ from $-3x$.
$8x - 3$. Power rule: $8x$ from $4x^2$ and $-3$ from $-3x$.
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What is the derivative of $h(x) = 5x^3 - 4x + 8$?
What is the derivative of $h(x) = 5x^3 - 4x + 8$?
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$15x^2 - 4$. Power rule: $15x^2$ from $5x^3$ and $-4$ from $-4x$.
$15x^2 - 4$. Power rule: $15x^2$ from $5x^3$ and $-4$ from $-4x$.
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Find the rate of change of the area of a square with side length $s$.
Find the rate of change of the area of a square with side length $s$.
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$2s$. Derivative of area $s^2$ gives rate of change.
$2s$. Derivative of area $s^2$ gives rate of change.
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What is the derivative of $f(x) = x^4 + 2x^2 + 1$?
What is the derivative of $f(x) = x^4 + 2x^2 + 1$?
Tap to reveal answer
$4x^3 + 4x$. Power rule: $4x^3$ from $x^4$ and $4x$ from $2x^2$.
$4x^3 + 4x$. Power rule: $4x^3$ from $x^4$ and $4x$ from $2x^2$.
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