Derivative Notation - AP Calculus AB
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What is the limit definition of the derivative of $f(x)$?
What is the limit definition of the derivative of $f(x)$?
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$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. Standard limit definition using difference quotient as $h$ approaches zero.
$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. Standard limit definition using difference quotient as $h$ approaches zero.
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Identify the derivative of $f(x) = \text{e}^{-x}$.
Identify the derivative of $f(x) = \text{e}^{-x}$.
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$f'(x) = -\text{e}^{-x}$. Chain rule with exponential function and negative exponent.
$f'(x) = -\text{e}^{-x}$. Chain rule with exponential function and negative exponent.
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State the derivative of $f(x)$ in prime notation.
State the derivative of $f(x)$ in prime notation.
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$f'(x)$. Prime notation for first derivative of function $f$.
$f'(x)$. Prime notation for first derivative of function $f$.
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Find the derivative of $f(x) = 3x^2 + 5x - 4$.
Find the derivative of $f(x) = 3x^2 + 5x - 4$.
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$f'(x) = 6x + 5$. Apply power rule to each term separately.
$f'(x) = 6x + 5$. Apply power rule to each term separately.
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How is the derivative of $y$ with respect to $x$ written using Leibniz's notation?
How is the derivative of $y$ with respect to $x$ written using Leibniz's notation?
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$\frac{dy}{dx}$. Standard Leibniz differential notation for derivatives.
$\frac{dy}{dx}$. Standard Leibniz differential notation for derivatives.
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What is the derivative of $\text{ln}(x)$?
What is the derivative of $\text{ln}(x)$?
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$\frac{1}{x}$. Natural logarithm derivative is reciprocal function.
$\frac{1}{x}$. Natural logarithm derivative is reciprocal function.
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What does $D_x[f(x)]$ represent?
What does $D_x[f(x)]$ represent?
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The derivative of $f(x)$ with respect to $x$. Operator notation where $D_x$ indicates differentiation with respect to $x$.
The derivative of $f(x)$ with respect to $x$. Operator notation where $D_x$ indicates differentiation with respect to $x$.
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What is the derivative of $\text{csc}(x)$?
What is the derivative of $\text{csc}(x)$?
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$ -\text{csc}(x)\text{cot}(x) $. Derivative of cosecant is negative cosecant cotangent.
$ -\text{csc}(x)\text{cot}(x) $. Derivative of cosecant is negative cosecant cotangent.
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What is the derivative of a difference $f(x) - g(x)$?
What is the derivative of a difference $f(x) - g(x)$?
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$f'(x) - g'(x)$. Derivative of difference equals difference of derivatives.
$f'(x) - g'(x)$. Derivative of difference equals difference of derivatives.
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What is the derivative of a sum $f(x) + g(x)$?
What is the derivative of a sum $f(x) + g(x)$?
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$f'(x) + g'(x)$. Derivative of sum equals sum of derivatives.
$f'(x) + g'(x)$. Derivative of sum equals sum of derivatives.
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What is the derivative of a constant function $c$?
What is the derivative of a constant function $c$?
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$0$. Constants have zero rate of change.
$0$. Constants have zero rate of change.
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What is the derivative of $\text{log}_a(x)$?
What is the derivative of $\text{log}_a(x)$?
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$\frac{1}{x\text{ln}(a)}$. Logarithm base $a$ derivative involves natural log of base.
$\frac{1}{x\text{ln}(a)}$. Logarithm base $a$ derivative involves natural log of base.
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What is the derivative of a product $f(x)g(x)$?
What is the derivative of a product $f(x)g(x)$?
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$f'(x)g(x) + f(x)g'(x)$. Product rule for differentiating products of functions.
$f'(x)g(x) + f(x)g'(x)$. Product rule for differentiating products of functions.
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State the derivative of $f(x) = x\text{ln}(x)$.
State the derivative of $f(x) = x\text{ln}(x)$.
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$f'(x) = 1 + \text{ln}(x)$. Product rule applied to $x$ and $\ln(x)$.
$f'(x) = 1 + \text{ln}(x)$. Product rule applied to $x$ and $\ln(x)$.
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What does $f'(x)$ represent graphically?
What does $f'(x)$ represent graphically?
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The slope of the tangent line to $f(x)$ at $x$. Derivative gives instantaneous rate of change at each point.
The slope of the tangent line to $f(x)$ at $x$. Derivative gives instantaneous rate of change at each point.
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What is the derivative of a quotient $\frac{f(x)}{g(x)}$?
What is the derivative of a quotient $\frac{f(x)}{g(x)}$?
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$\frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$. Quotient rule for differentiating ratios of functions.
$\frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$. Quotient rule for differentiating ratios of functions.
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What is the derivative of $f(x) = 3^x$?
What is the derivative of $f(x) = 3^x$?
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$f'(x) = 3^x\text{ln}(3)$. Exponential with base $a$ involves $\ln(a)$ factor.
$f'(x) = 3^x\text{ln}(3)$. Exponential with base $a$ involves $\ln(a)$ factor.
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What is the definition of the derivative of a function at a point $x=a$?
What is the definition of the derivative of a function at a point $x=a$?
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$f'(a) = \frac{d}{dx}f(x) \bigg|_{x=a}$. Notation shows derivative of $f$ evaluated at point $a$.
$f'(a) = \frac{d}{dx}f(x) \bigg|_{x=a}$. Notation shows derivative of $f$ evaluated at point $a$.
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Differentiate $f(x) = \frac{x^2}{x+1}$.
Differentiate $f(x) = \frac{x^2}{x+1}$.
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$f'(x) = \frac{x(x+2)}{(x+1)^2}$. Apply quotient rule to rational function.
$f'(x) = \frac{x(x+2)}{(x+1)^2}$. Apply quotient rule to rational function.
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State the Power Rule for derivatives.
State the Power Rule for derivatives.
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$\frac{d}{dx}x^n = nx^{n-1}$. Fundamental derivative rule for power functions.
$\frac{d}{dx}x^n = nx^{n-1}$. Fundamental derivative rule for power functions.
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Find the derivative of $f(x) = \text{e}^{2x}$.
Find the derivative of $f(x) = \text{e}^{2x}$.
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$f'(x) = 2\text{e}^{2x}$. Chain rule: derivative of outer times derivative of inner.
$f'(x) = 2\text{e}^{2x}$. Chain rule: derivative of outer times derivative of inner.
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What is the Chain Rule in derivative notation?
What is the Chain Rule in derivative notation?
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$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Formula for differentiating composite functions.
$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Formula for differentiating composite functions.
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Differentiate $f(x) = 5\text{sin}(x)$.
Differentiate $f(x) = 5\text{sin}(x)$.
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$f'(x) = 5\text{cos}(x)$. Constant factor rule: $\frac{d}{dx}[cf(x)] = cf'(x)$.
$f'(x) = 5\text{cos}(x)$. Constant factor rule: $\frac{d}{dx}[cf(x)] = cf'(x)$.
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Find the derivative of $f(x) = \text{ln}(x^2)$.
Find the derivative of $f(x) = \text{ln}(x^2)$.
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$f'(x) = \frac{2}{x}$. Use property $\ln(x^2) = 2\ln(x)$ then differentiate.
$f'(x) = \frac{2}{x}$. Use property $\ln(x^2) = 2\ln(x)$ then differentiate.
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What is the derivative of $f(x) = \text{sin}^2(x)$?
What is the derivative of $f(x) = \text{sin}^2(x)$?
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$f'(x) = 2\text{sin}(x)\text{cos}(x)$. Chain rule with power of sine function.
$f'(x) = 2\text{sin}(x)\text{cos}(x)$. Chain rule with power of sine function.
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Identify the derivative of $\frac{1}{x}$.
Identify the derivative of $\frac{1}{x}$.
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$-\frac{1}{x^2}$. Rewrite as $x^{-1}$ and apply power rule.
$-\frac{1}{x^2}$. Rewrite as $x^{-1}$ and apply power rule.
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What is the derivative of $\text{sin}(x)$?
What is the derivative of $\text{sin}(x)$?
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$\text{cos}(x)$. Derivative of sine is cosine.
$\text{cos}(x)$. Derivative of sine is cosine.
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Identify the derivative notation for $y$ with respect to $x$.
Identify the derivative notation for $y$ with respect to $x$.
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$\frac{dy}{dx}$. Leibniz notation for derivative of $y$ with respect to $x$.
$\frac{dy}{dx}$. Leibniz notation for derivative of $y$ with respect to $x$.
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What is the derivative of $\text{sec}(x)$?
What is the derivative of $\text{sec}(x)$?
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$\text{sec}(x)\text{tan}(x)$. Derivative of secant involves secant times tangent.
$\text{sec}(x)\text{tan}(x)$. Derivative of secant involves secant times tangent.
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What is the derivative of $\text{sec}(x)$?
What is the derivative of $\text{sec}(x)$?
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$\text{sec}(x)\text{tan}(x)$. Derivative of secant involves secant times tangent.
$\text{sec}(x)\text{tan}(x)$. Derivative of secant involves secant times tangent.
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