Rates of Change in Applied Concepts - AP Calculus BC
Card 1 of 30
Determine the derivative of $f(x) = \cos(x)$.
Determine the derivative of $f(x) = \cos(x)$.
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$f'(x) = -\sin(x)$. Cosine derivative is negative sine.
$f'(x) = -\sin(x)$. Cosine derivative is negative sine.
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Determine $\frac{d}{dx}(5x - 1)$ at $x = 1$.
Determine $\frac{d}{dx}(5x - 1)$ at $x = 1$.
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$5$, so the rate is $5$. Linear function has constant derivative.
$5$, so the rate is $5$. Linear function has constant derivative.
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Find the rate of change of the volume of a sphere with respect to its radius.
Find the rate of change of the volume of a sphere with respect to its radius.
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$\frac{dV}{dr} = 4\pi r^2$. Derivative of $V = \frac{4}{3}\pi r^3$ using power rule.
$\frac{dV}{dr} = 4\pi r^2$. Derivative of $V = \frac{4}{3}\pi r^3$ using power rule.
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Find the derivative of $f(x) = \arcsin(x)$.
Find the derivative of $f(x) = \arcsin(x)$.
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$f'(x) = \frac{1}{\sqrt{1-x^2}}$. Arcsine derivative formula.
$f'(x) = \frac{1}{\sqrt{1-x^2}}$. Arcsine derivative formula.
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Determine $\frac{d}{dx}(x^2 + 3x + 2)$.
Determine $\frac{d}{dx}(x^2 + 3x + 2)$.
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$2x + 3$. Sum rule and power rule applied term by term.
$2x + 3$. Sum rule and power rule applied term by term.
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What is the derivative of $f(x) = \cot(x)$?
What is the derivative of $f(x) = \cot(x)$?
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$f'(x) = -\csc^2(x)$. Cotangent derivative is negative cosecant squared.
$f'(x) = -\csc^2(x)$. Cotangent derivative is negative cosecant squared.
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What is the rate of change of the area of a circle with respect to its radius?
What is the rate of change of the area of a circle with respect to its radius?
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$\frac{dA}{dr} = 2\pi r$. Derivative of $A = \pi r^2$ using power rule.
$\frac{dA}{dr} = 2\pi r$. Derivative of $A = \pi r^2$ using power rule.
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Calculate the derivative of $f(x) = \sin(x)$.
Calculate the derivative of $f(x) = \sin(x)$.
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$f'(x) = \cos(x)$. Sine derivative is cosine.
$f'(x) = \cos(x)$. Sine derivative is cosine.
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Evaluate $\frac{d}{dx}(x^2 + x)$ at $x = 2$.
Evaluate $\frac{d}{dx}(x^2 + x)$ at $x = 2$.
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$2x + 1$, so $f'(2) = 5$. Sum rule applied, then evaluate at $x = 2$.
$2x + 1$, so $f'(2) = 5$. Sum rule applied, then evaluate at $x = 2$.
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What is the derivative of $f(x) = \tan(x)$?
What is the derivative of $f(x) = \tan(x)$?
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$f'(x) = \sec^2(x)$. Tangent derivative is secant squared.
$f'(x) = \sec^2(x)$. Tangent derivative is secant squared.
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State the product rule for the derivative of $u(x)v(x)$.
State the product rule for the derivative of $u(x)v(x)$.
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$(uv)' = u'v + uv'$. Product rule for multiplied functions.
$(uv)' = u'v + uv'$. Product rule for multiplied functions.
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Find the derivative of $f(x) = \sec(x)$.
Find the derivative of $f(x) = \sec(x)$.
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$f'(x) = \sec(x)\tan(x)$. Secant derivative formula.
$f'(x) = \sec(x)\tan(x)$. Secant derivative formula.
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Calculate the derivative of $f(x) = \ln(ax)$ where $a$ is a constant.
Calculate the derivative of $f(x) = \ln(ax)$ where $a$ is a constant.
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$f'(x) = \frac{1}{x}$. Constant factor $a$ cancels in logarithm derivative.
$f'(x) = \frac{1}{x}$. Constant factor $a$ cancels in logarithm derivative.
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What is the rate of change of $y = 2x^3$ at $x = 2$?
What is the rate of change of $y = 2x^3$ at $x = 2$?
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$f'(x) = 6x^2$, so $f'(2) = 24$. Constant multiple rule with power rule.
$f'(x) = 6x^2$, so $f'(2) = 24$. Constant multiple rule with power rule.
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Find the rate of change of $f(x) = x^3$ at $x = 4$.
Find the rate of change of $f(x) = x^3$ at $x = 4$.
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$f'(x) = 3x^2$, so $f'(4) = 48$. Power rule gives $3x^2$, substitute $x = 4$.
$f'(x) = 3x^2$, so $f'(4) = 48$. Power rule gives $3x^2$, substitute $x = 4$.
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Calculate the derivative of $f(x) = 2x^3 - 3x^2 + x$.
Calculate the derivative of $f(x) = 2x^3 - 3x^2 + x$.
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$6x^2 - 6x + 1$. Power rule applied to polynomial terms.
$6x^2 - 6x + 1$. Power rule applied to polynomial terms.
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Find the rate of change for $f(x) = 4x^2$ at $x = 3$.
Find the rate of change for $f(x) = 4x^2$ at $x = 3$.
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$f'(x) = 8x$, so $f'(3) = 24$. Power rule gives $8x$, substitute $x = 3$.
$f'(x) = 8x$, so $f'(3) = 24$. Power rule gives $8x$, substitute $x = 3$.
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Evaluate $\frac{d}{dx}(x^3 - 2x + 4)$ at $x = 0$.
Evaluate $\frac{d}{dx}(x^3 - 2x + 4)$ at $x = 0$.
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$3x^2 - 2$, so $f'(0) = -2$. Differentiate then evaluate at $x = 0$.
$3x^2 - 2$, so $f'(0) = -2$. Differentiate then evaluate at $x = 0$.
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Calculate the derivative of $f(x) = 5x^2 - 3x + 7$.
Calculate the derivative of $f(x) = 5x^2 - 3x + 7$.
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$10x - 3$. Power rule applied to polynomial.
$10x - 3$. Power rule applied to polynomial.
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Identify the rate of change of $y = x^3$ at $x = 2$.
Identify the rate of change of $y = x^3$ at $x = 2$.
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$f'(x) = 3x^2$, so $f'(2) = 12$. Apply power rule then substitute $x = 2$.
$f'(x) = 3x^2$, so $f'(2) = 12$. Apply power rule then substitute $x = 2$.
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Find the derivative of $f(x) = e^x$.
Find the derivative of $f(x) = e^x$.
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$f'(x) = e^x$. Exponential function derivative equals itself.
$f'(x) = e^x$. Exponential function derivative equals itself.
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What is the derivative of $f(x) = \arccos(x)$?
What is the derivative of $f(x) = \arccos(x)$?
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$f'(x) = -\frac{1}{\sqrt{1-x^2}}$. Arccosine derivative is negative of arcsine.
$f'(x) = -\frac{1}{\sqrt{1-x^2}}$. Arccosine derivative is negative of arcsine.
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Determine the derivative of $f(x) = \arctan(x)$.
Determine the derivative of $f(x) = \arctan(x)$.
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$f'(x) = \frac{1}{1+x^2}$. Arctangent derivative formula.
$f'(x) = \frac{1}{1+x^2}$. Arctangent derivative formula.
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State the Chain Rule formula for derivatives.
State the Chain Rule formula for derivatives.
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$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$. Composite function differentiation rule.
$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$. Composite function differentiation rule.
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What is the formula for the rate of change of a function $f(x)$?
What is the formula for the rate of change of a function $f(x)$?
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$f'(x)$. The derivative represents instantaneous rate of change.
$f'(x)$. The derivative represents instantaneous rate of change.
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Calculate the rate of change of $f(x) = x^2 + 2x$ at $x = 1$.
Calculate the rate of change of $f(x) = x^2 + 2x$ at $x = 1$.
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$f'(x) = 2x + 2$, so $f'(1) = 4$. Apply power and sum rules, then evaluate.
$f'(x) = 2x + 2$, so $f'(1) = 4$. Apply power and sum rules, then evaluate.
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Determine the rate of change for $f(x) = x^2$ at $x = 5$.
Determine the rate of change for $f(x) = x^2$ at $x = 5$.
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$f'(x) = 2x$, so $f'(5) = 10$. Power rule gives $2x$, substitute $x = 5$.
$f'(x) = 2x$, so $f'(5) = 10$. Power rule gives $2x$, substitute $x = 5$.
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Calculate the derivative of $f(x) = x^{-1}$.
Calculate the derivative of $f(x) = x^{-1}$.
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$f'(x) = -x^{-2}$. Power rule applied to negative exponent.
$f'(x) = -x^{-2}$. Power rule applied to negative exponent.
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Find $\frac{d}{dx} (3x^3 - 5x^2 + 4)$.
Find $\frac{d}{dx} (3x^3 - 5x^2 + 4)$.
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$9x^2 - 10x$. Power rule applied to each term.
$9x^2 - 10x$. Power rule applied to each term.
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Evaluate the derivative of $f(x) = \frac{1}{x}$ at $x = 1$.
Evaluate the derivative of $f(x) = \frac{1}{x}$ at $x = 1$.
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$-\frac{1}{x^2}$, so $f'(1) = -1$. Reciprocal function derivative using power rule.
$-\frac{1}{x^2}$, so $f'(1) = -1$. Reciprocal function derivative using power rule.
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