Introducing Calculus - AP Calculus BC
Card 1 of 30
State the formula for the derivative of $f(x)$ using limits.
State the formula for the derivative of $f(x)$ using limits.
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$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. This is the formal limit definition of the derivative.
$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. This is the formal limit definition of the derivative.
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Identify the derivative of $f(x) = \text{cos}(3x)$.
Identify the derivative of $f(x) = \text{cos}(3x)$.
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$f'(x) = -3 \text{sin}(3x)$. Apply Chain Rule: derivative of $\cos(u)$ is $-\sin(u) \cdot u'$ where $u = 3x$.
$f'(x) = -3 \text{sin}(3x)$. Apply Chain Rule: derivative of $\cos(u)$ is $-\sin(u) \cdot u'$ where $u = 3x$.
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Which rule is used for differentiating compositions of functions?
Which rule is used for differentiating compositions of functions?
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The Chain Rule. Use this rule for composite functions like $f(g(x))$.
The Chain Rule. Use this rule for composite functions like $f(g(x))$.
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Find the derivative of $f(x) = \frac{1}{x}$.
Find the derivative of $f(x) = \frac{1}{x}$.
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$f'(x) = -\frac{1}{x^2}$. Rewrite as $x^{-1}$ and apply the Power Rule.
$f'(x) = -\frac{1}{x^2}$. Rewrite as $x^{-1}$ and apply the Power Rule.
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Find the derivative of $f(x) = \text{tan}(x^2)$.
Find the derivative of $f(x) = \text{tan}(x^2)$.
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$f'(x) = 2x \text{sec}^2(x^2)$. Apply Chain Rule: derivative of $\tan(u)$ is $\sec^2(u) \cdot u'$ where $u = x^2$.
$f'(x) = 2x \text{sec}^2(x^2)$. Apply Chain Rule: derivative of $\tan(u)$ is $\sec^2(u) \cdot u'$ where $u = x^2$.
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What is the geometric interpretation of a derivative?
What is the geometric interpretation of a derivative?
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The slope of the tangent line to the curve at a point. The derivative gives the instantaneous slope at any point.
The slope of the tangent line to the curve at a point. The derivative gives the instantaneous slope at any point.
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Which function's derivative is $e^x$?
Which function's derivative is $e^x$?
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$e^x$. The exponential function $e^x$ is its own derivative.
$e^x$. The exponential function $e^x$ is its own derivative.
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Find the derivative of $f(x) = x^4 - 2x^2 + x$.
Find the derivative of $f(x) = x^4 - 2x^2 + x$.
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$f'(x) = 4x^3 - 4x + 1$. Apply Power Rule to each term: $4x^3 - 4x + 1$.
$f'(x) = 4x^3 - 4x + 1$. Apply Power Rule to each term: $4x^3 - 4x + 1$.
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Identify the derivative of a constant function.
Identify the derivative of a constant function.
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Zero. Constants have no rate of change, so their derivative is zero.
Zero. Constants have no rate of change, so their derivative is zero.
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Find the derivative of $f(x) = \text{ln}(x)$.
Find the derivative of $f(x) = \text{ln}(x)$.
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$f'(x) = \frac{1}{x}$. The natural logarithm's derivative is the reciprocal function.
$f'(x) = \frac{1}{x}$. The natural logarithm's derivative is the reciprocal function.
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Which rule is used to find the derivative of a product of two functions?
Which rule is used to find the derivative of a product of two functions?
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The Product Rule. Use this when differentiating products of two functions.
The Product Rule. Use this when differentiating products of two functions.
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What is the derivative of $\text{arccos}(x)$?
What is the derivative of $\text{arccos}(x)$?
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$-\frac{1}{\sqrt{1-x^2}}$. This is the standard derivative formula for inverse cosine.
$-\frac{1}{\sqrt{1-x^2}}$. This is the standard derivative formula for inverse cosine.
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What is the derivative of $\text{ln}(f(x))$?
What is the derivative of $\text{ln}(f(x))$?
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$\frac{f'(x)}{f(x)}$. Use the Chain Rule with the natural logarithm function.
$\frac{f'(x)}{f(x)}$. Use the Chain Rule with the natural logarithm function.
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What is the derivative of $\text{arcsin}(x)$?
What is the derivative of $\text{arcsin}(x)$?
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$\frac{1}{\sqrt{1-x^2}}$. This is the standard derivative formula for inverse sine.
$\frac{1}{\sqrt{1-x^2}}$. This is the standard derivative formula for inverse sine.
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Find the derivative of $f(x) = \text{e}^{x^2}$.
Find the derivative of $f(x) = \text{e}^{x^2}$.
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$f'(x) = 2x \text{e}^{x^2}$. Apply Chain Rule: derivative of $e^u$ is $e^u \cdot u'$ where $u = x^2$.
$f'(x) = 2x \text{e}^{x^2}$. Apply Chain Rule: derivative of $e^u$ is $e^u \cdot u'$ where $u = x^2$.
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What is the derivative of $\cos(x)$?
What is the derivative of $\cos(x)$?
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-$\sin(x)$. The derivative of cosine is negative sine.
-$\sin(x)$. The derivative of cosine is negative sine.
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What is the derivative of $\text{sec}(x)$?
What is the derivative of $\text{sec}(x)$?
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$\text{sec}(x)\text{tan}(x)$. This is the standard derivative formula for secant.
$\text{sec}(x)\text{tan}(x)$. This is the standard derivative formula for secant.
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Find the derivative of $f(x) = x \text{e}^x$ using the Product Rule.
Find the derivative of $f(x) = x \text{e}^x$ using the Product Rule.
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$f'(x) = \text{e}^x + x \text{e}^x$. Apply Product Rule: $1 \cdot e^x + x \cdot e^x = e^x(1 + x)$.
$f'(x) = \text{e}^x + x \text{e}^x$. Apply Product Rule: $1 \cdot e^x + x \cdot e^x = e^x(1 + x)$.
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What is the derivative of $\text{cot}(x)$?
What is the derivative of $\text{cot}(x)$?
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$-\text{csc}^2(x)$. This is the standard derivative formula for cotangent.
$-\text{csc}^2(x)$. This is the standard derivative formula for cotangent.
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Find the derivative of $f(x) = \text{ln}(x^2 + 1)$.
Find the derivative of $f(x) = \text{ln}(x^2 + 1)$.
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$f'(x) = \frac{2x}{x^2 + 1}$. Apply Chain Rule: $\frac{1}{x^2 + 1} \cdot 2x$.
$f'(x) = \frac{2x}{x^2 + 1}$. Apply Chain Rule: $\frac{1}{x^2 + 1} \cdot 2x$.
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What is the derivative of $\text{csc}(x)$?
What is the derivative of $\text{csc}(x)$?
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$-\text{csc}(x)\text{cot}(x)$. This is the standard derivative formula for cosecant.
$-\text{csc}(x)\text{cot}(x)$. This is the standard derivative formula for cosecant.
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Identify the derivative of $f(x) = \frac{1}{\text{sin}(x)}$.
Identify the derivative of $f(x) = \frac{1}{\text{sin}(x)}$.
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$f'(x) = -\frac{\text{cos}(x)}{\text{sin}^2(x)}$. Rewrite as $\csc(x)$ and use the derivative formula.
$f'(x) = -\frac{\text{cos}(x)}{\text{sin}^2(x)}$. Rewrite as $\csc(x)$ and use the derivative formula.
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State the derivative of $\text{arctan}(x)$.
State the derivative of $\text{arctan}(x)$.
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$\frac{1}{1+x^2}$. This is the standard derivative formula for inverse tangent.
$\frac{1}{1+x^2}$. This is the standard derivative formula for inverse tangent.
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What is the derivative of $\text{e}^{f(x)}$?
What is the derivative of $\text{e}^{f(x)}$?
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$f'(x) \text{e}^{f(x)}$. Use the Chain Rule with the exponential function.
$f'(x) \text{e}^{f(x)}$. Use the Chain Rule with the exponential function.
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Identify the derivative of the inverse function $f^{-1}(x)$.
Identify the derivative of the inverse function $f^{-1}(x)$.
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$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$. The derivative of an inverse function uses this reciprocal relationship.
$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$. The derivative of an inverse function uses this reciprocal relationship.
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Which rule is used to find the derivative of a quotient of two functions?
Which rule is used to find the derivative of a quotient of two functions?
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The Quotient Rule. Use this when differentiating quotients of two functions.
The Quotient Rule. Use this when differentiating quotients of two functions.
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State the Product Rule for derivatives.
State the Product Rule for derivatives.
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If $u(x)$ and $v(x)$, then $(uv)' = u'v + uv'$. First function times derivative of second plus second times derivative of first.
If $u(x)$ and $v(x)$, then $(uv)' = u'v + uv'$. First function times derivative of second plus second times derivative of first.
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State the Power Rule for derivatives.
State the Power Rule for derivatives.
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If $f(x) = x^n$, then $f'(x) = nx^{n-1}$. Multiply by the exponent, then reduce the exponent by 1.
If $f(x) = x^n$, then $f'(x) = nx^{n-1}$. Multiply by the exponent, then reduce the exponent by 1.
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Find the derivative of $f(x) = 3x^2 + 2x + 1$.
Find the derivative of $f(x) = 3x^2 + 2x + 1$.
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$f'(x) = 6x + 2$. Apply the Power Rule to each term separately.
$f'(x) = 6x + 2$. Apply the Power Rule to each term separately.
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What is the definition of a derivative?
What is the definition of a derivative?
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The limit of the average rate of change as the interval approaches zero. This captures the instantaneous rate of change concept.
The limit of the average rate of change as the interval approaches zero. This captures the instantaneous rate of change concept.
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