Derivative Rules of Constant, Sum, Difference - AP Calculus AB
Card 1 of 30
Given $f(x) = x^2 + 3x$, find $f'(x)$.
Given $f(x) = x^2 + 3x$, find $f'(x)$.
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2x + 3. Apply power rule to each term separately.
2x + 3. Apply power rule to each term separately.
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What is the derivative of a constant $c$ with respect to $x$?
What is the derivative of a constant $c$ with respect to $x$?
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- Constants have zero rate of change.
- Constants have zero rate of change.
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Calculate the derivative of $7x^3 - 4x$.
Calculate the derivative of $7x^3 - 4x$.
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21x^2 - 4. Use power rule: $7 \cdot 3x^2 - 4 = 21x^2 - 4$.
21x^2 - 4. Use power rule: $7 \cdot 3x^2 - 4 = 21x^2 - 4$.
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State the derivative of the function $-5$.
State the derivative of the function $-5$.
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- Any constant has derivative zero.
- Any constant has derivative zero.
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Calculate the derivative of $12x$.
Calculate the derivative of $12x$.
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- Derivative of linear term $12x$ is the coefficient.
- Derivative of linear term $12x$ is the coefficient.
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What is the derivative of a constant multiplied by $x$, $7x$?
What is the derivative of a constant multiplied by $x$, $7x$?
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- Constant multiple rule: derivative of $cx$ is $c$.
- Constant multiple rule: derivative of $cx$ is $c$.
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Find the derivative of $8x - 2$.
Find the derivative of $8x - 2$.
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- Derivative of linear term is its coefficient.
- Derivative of linear term is its coefficient.
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State the derivative of $x^n$ where $n$ is a constant.
State the derivative of $x^n$ where $n$ is a constant.
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$nx^{n-1}$. Power rule: bring down exponent, reduce by 1.
$nx^{n-1}$. Power rule: bring down exponent, reduce by 1.
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Given $f(x) = x^2 + 3x$, find $f'(x)$.
Given $f(x) = x^2 + 3x$, find $f'(x)$.
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2x + 3. Apply power rule to each term separately.
2x + 3. Apply power rule to each term separately.
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What is the derivative of $cf(x)$, where $c$ is a constant?
What is the derivative of $cf(x)$, where $c$ is a constant?
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$(cf)' = cf'$. Constant multiple rule: factor out the constant.
$(cf)' = cf'$. Constant multiple rule: factor out the constant.
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State the derivative rule for the difference of two functions $f(x)$ and $g(x)$.
State the derivative rule for the difference of two functions $f(x)$ and $g(x)$.
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$(f - g)' = f' - g'$. Derivative of difference equals difference of derivatives.
$(f - g)' = f' - g'$. Derivative of difference equals difference of derivatives.
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What is the derivative of a constant $c$ with respect to $x$?
What is the derivative of a constant $c$ with respect to $x$?
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- Constants have zero rate of change.
- Constants have zero rate of change.
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Calculate the derivative of $5x - 3$.
Calculate the derivative of $5x - 3$.
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- Derivative of linear function is the coefficient of $x$.
- Derivative of linear function is the coefficient of $x$.
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State the derivative rule for the sum of two functions $f(x)$ and $g(x)$.
State the derivative rule for the sum of two functions $f(x)$ and $g(x)$.
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$(f + g)' = f' + g'$. Derivative of sum equals sum of derivatives.
$(f + g)' = f' + g'$. Derivative of sum equals sum of derivatives.
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Calculate the derivative of $7x^3 - 4x$.
Calculate the derivative of $7x^3 - 4x$.
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21x^2 - 4. Use power rule: $7 \cdot 3x^2 - 4 = 21x^2 - 4$.
21x^2 - 4. Use power rule: $7 \cdot 3x^2 - 4 = 21x^2 - 4$.
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Find the derivative of $8x - 2$.
Find the derivative of $8x - 2$.
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- Derivative of linear term is its coefficient.
- Derivative of linear term is its coefficient.
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What is $ rac{d}{dx}[2f(x)+3g(x)-5]$?
What is $ rac{d}{dx}[2f(x)+3g(x)-5]$?
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$2f'(x)+3g'(x)$. Constant -5 has derivative 0; apply rules to rest.
$2f'(x)+3g'(x)$. Constant -5 has derivative 0; apply rules to rest.
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What is $ rac{d}{dx}[f(x)-g(x)+h(x)]$?
What is $ rac{d}{dx}[f(x)-g(x)+h(x)]$?
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$f'(x)-g'(x)+h'(x)$. Apply sum/difference rules term by term.
$f'(x)-g'(x)+h'(x)$. Apply sum/difference rules term by term.
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What is $ rac{d}{dx}[f(x)+g(x)+h(x)]$?
What is $ rac{d}{dx}[f(x)+g(x)+h(x)]$?
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$f'(x)+g'(x)+h'(x)$. Sum rule extends to any number of functions.
$f'(x)+g'(x)+h'(x)$. Sum rule extends to any number of functions.
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What is $ rac{d}{dx}[12-g(x)]$?
What is $ rac{d}{dx}[12-g(x)]$?
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$-g'(x)$. Derivative of 12 is 0; $-g(x)$ gives $-g'(x)$.
$-g'(x)$. Derivative of 12 is 0; $-g(x)$ gives $-g'(x)$.
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What is $ rac{d}{dx}[f(x)+9]$?
What is $ rac{d}{dx}[f(x)+9]$?
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$f'(x)$. Derivative of constant 9 is 0, leaving $f'(x)$.
$f'(x)$. Derivative of constant 9 is 0, leaving $f'(x)$.
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What is $ rac{d}{dx}[-2g(x)]$?
What is $ rac{d}{dx}[-2g(x)]$?
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$-2g'(x)$. Apply constant multiple rule: pull out -2.
$-2g'(x)$. Apply constant multiple rule: pull out -2.
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What is $ rac{d}{dx}[5f(x)]$?
What is $ rac{d}{dx}[5f(x)]$?
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$5f'(x)$. Apply constant multiple rule: pull out 5.
$5f'(x)$. Apply constant multiple rule: pull out 5.
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What is $ rac{d}{dx}[-3]$?
What is $ rac{d}{dx}[-3]$?
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$0$. -3 is a constant, so its derivative is 0.
$0$. -3 is a constant, so its derivative is 0.
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What is $ rac{d}{dx}[7]$?
What is $ rac{d}{dx}[7]$?
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$0$. 7 is a constant, so its derivative is 0.
$0$. 7 is a constant, so its derivative is 0.
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State the difference rule: What is $ rac{d}{dx}[f(x)-g(x)]$?
State the difference rule: What is $ rac{d}{dx}[f(x)-g(x)]$?
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$f'(x)-g'(x)$. Differentiate each term separately and subtract.
$f'(x)-g'(x)$. Differentiate each term separately and subtract.
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State the sum rule: What is $ rac{d}{dx}[f(x)+g(x)]$?
State the sum rule: What is $ rac{d}{dx}[f(x)+g(x)]$?
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$f'(x)+g'(x)$. Differentiate each term separately and add.
$f'(x)+g'(x)$. Differentiate each term separately and add.
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What is $ rac{d}{dx}[3f(x)-2g(x)]$?
What is $ rac{d}{dx}[3f(x)-2g(x)]$?
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$3f'(x)-2g'(x)$. Apply constant multiple rule to each term.
$3f'(x)-2g'(x)$. Apply constant multiple rule to each term.
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State the constant multiple rule: What is $ rac{d}{dx}[c ext{ }f(x)]$ for constant $c$?
State the constant multiple rule: What is $ rac{d}{dx}[c ext{ }f(x)]$ for constant $c$?
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$c ext{ }f'(x)$. Pull out the constant and multiply by the derivative.
$c ext{ }f'(x)$. Pull out the constant and multiply by the derivative.
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State the constant rule: What is $ rac{d}{dx}[c]$ for a constant $c$?
State the constant rule: What is $ rac{d}{dx}[c]$ for a constant $c$?
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$0$. The derivative of any constant is zero.
$0$. The derivative of any constant is zero.
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