Derivatives of Trigonometry and Logarithmic Functions - AP Calculus BC
Card 1 of 30
Find the second derivative of $y = \text{cos } x$.
Find the second derivative of $y = \text{cos } x$.
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$\frac{d^2y}{dx^2} = -\text{cos } x$. Differentiate $f'(x) = -\text{sin } x$ to get $f''(x) = -\text{cos } x$.
$\frac{d^2y}{dx^2} = -\text{cos } x$. Differentiate $f'(x) = -\text{sin } x$ to get $f''(x) = -\text{cos } x$.
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Determine the derivative of $f(x) = x\text{ln } x$.
Determine the derivative of $f(x) = x\text{ln } x$.
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$f'(x) = \text{ln } x + 1$. Use product rule: $(uv)' = u'v + uv'$.
$f'(x) = \text{ln } x + 1$. Use product rule: $(uv)' = u'v + uv'$.
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Determine $f'(x)$ if $f(x) = \text{cos } x + \text{sin } x$.
Determine $f'(x)$ if $f(x) = \text{cos } x + \text{sin } x$.
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$f'(x) = -\text{sin } x + \text{cos } x$. Use sum rule: derivative of each term separately.
$f'(x) = -\text{sin } x + \text{cos } x$. Use sum rule: derivative of each term separately.
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Evaluate $f'(x)$ for $f(x) = \text{cos } x - e^x$ at $x = 0$.
Evaluate $f'(x)$ for $f(x) = \text{cos } x - e^x$ at $x = 0$.
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$f'(0) = -1$. $f'(x) = -\text{sin } x - e^x$, so $f'(0) = 0 - 1 = -1$.
$f'(0) = -1$. $f'(x) = -\text{sin } x - e^x$, so $f'(0) = 0 - 1 = -1$.
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What is the derivative of $\text{cos } x$?
What is the derivative of $\text{cos } x$?
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$-\text{sin } x$. Basic derivative rule for cosine function.
$-\text{sin } x$. Basic derivative rule for cosine function.
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Find the derivative: $y = \text{cos } x$.
Find the derivative: $y = \text{cos } x$.
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$\frac{dy}{dx} = -\text{sin } x$. Apply the derivative rule for $\text{cos } x$.
$\frac{dy}{dx} = -\text{sin } x$. Apply the derivative rule for $\text{cos } x$.
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Find the derivative of $f(x) = \sin(2x)$.
Find the derivative of $f(x) = \sin(2x)$.
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$f'(x) = 2\cos(2x)$. Use chain rule with inner function $2x$.
$f'(x) = 2\cos(2x)$. Use chain rule with inner function $2x$.
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Find the second derivative of $y = \text{ln } x$.
Find the second derivative of $y = \text{ln } x$.
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$\frac{d^2y}{dx^2} = -\frac{1}{x^2}$. Differentiate $f'(x) = \frac{1}{x}$ to get $f''(x) = -\frac{1}{x^2}$.
$\frac{d^2y}{dx^2} = -\frac{1}{x^2}$. Differentiate $f'(x) = \frac{1}{x}$ to get $f''(x) = -\frac{1}{x^2}$.
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Determine the derivative of $f(x) = x\text{sin } x$.
Determine the derivative of $f(x) = x\text{sin } x$.
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$f'(x) = \text{sin } x + x\text{cos } x$. Use product rule: $(uv)' = u'v + uv'$.
$f'(x) = \text{sin } x + x\text{cos } x$. Use product rule: $(uv)' = u'v + uv'$.
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Differentiate $f(x) = 3\text{cos } x - 2\text{sin } x$.
Differentiate $f(x) = 3\text{cos } x - 2\text{sin } x$.
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$f'(x) = -3\text{sin } x - 2\text{cos } x$. Use sum/difference rule with constant multiples.
$f'(x) = -3\text{sin } x - 2\text{cos } x$. Use sum/difference rule with constant multiples.
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Find the slope of the tangent to $y = \text{ln } x$ at $x = 4$.
Find the slope of the tangent to $y = \text{ln } x$ at $x = 4$.
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Slope = $\frac{1}{4}$. The slope equals the derivative at the point.
Slope = $\frac{1}{4}$. The slope equals the derivative at the point.
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Evaluate $f'(x)$ for $f(x) = e^x - \text{ln } x$ at $x = 1$.
Evaluate $f'(x)$ for $f(x) = e^x - \text{ln } x$ at $x = 1$.
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$f'(1) = e - 1$. $f'(x) = e^x - \frac{1}{x}$, so $f'(1) = e - 1$.
$f'(1) = e - 1$. $f'(x) = e^x - \frac{1}{x}$, so $f'(1) = e - 1$.
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Find the second derivative of $y = e^x$.
Find the second derivative of $y = e^x$.
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$\frac{d^2y}{dx^2} = e^x$. Differentiate $f'(x) = e^x$ to get $f''(x) = e^x$.
$\frac{d^2y}{dx^2} = e^x$. Differentiate $f'(x) = e^x$ to get $f''(x) = e^x$.
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Find the derivative of $f(x) = \text{ln }(5x)$.
Find the derivative of $f(x) = \text{ln }(5x)$.
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$f'(x) = \frac{1}{x}$. Constant multiple rule: $\frac{d}{dx}[\text{ln}(5x)] = \frac{1}{5x} \cdot 5 = \frac{1}{x}$.
$f'(x) = \frac{1}{x}$. Constant multiple rule: $\frac{d}{dx}[\text{ln}(5x)] = \frac{1}{5x} \cdot 5 = \frac{1}{x}$.
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Find the derivative of $f(x) = e^{2x}$.
Find the derivative of $f(x) = e^{2x}$.
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$f'(x) = 2e^{2x}$. Use chain rule with inner function $2x$.
$f'(x) = 2e^{2x}$. Use chain rule with inner function $2x$.
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Find the derivative of $f(x) = \text{cos }(3x)$.
Find the derivative of $f(x) = \text{cos }(3x)$.
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$f'(x) = -3\text{sin }(3x)$. Use chain rule with inner function $3x$.
$f'(x) = -3\text{sin }(3x)$. Use chain rule with inner function $3x$.
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Determine the derivative of $f(x) = xe^x$.
Determine the derivative of $f(x) = xe^x$.
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$f'(x) = e^x + xe^x$. Use product rule: $(uv)' = u'v + uv'$.
$f'(x) = e^x + xe^x$. Use product rule: $(uv)' = u'v + uv'$.
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Determine the derivative of $f(x) = x\text{cos } x$.
Determine the derivative of $f(x) = x\text{cos } x$.
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$f'(x) = \text{cos } x - x\text{sin } x$. Use product rule: $(uv)' = u'v + uv'$.
$f'(x) = \text{cos } x - x\text{sin } x$. Use product rule: $(uv)' = u'v + uv'$.
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Find the derivative: $y = \text{ln } x$.
Find the derivative: $y = \text{ln } x$.
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$\frac{dy}{dx} = \frac{1}{x}$. Apply the derivative rule for $\text{ln } x$.
$\frac{dy}{dx} = \frac{1}{x}$. Apply the derivative rule for $\text{ln } x$.
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Differentiate $f(x) = 5e^x + 7\text{ln } x$.
Differentiate $f(x) = 5e^x + 7\text{ln } x$.
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$f'(x) = 5e^x + \frac{7}{x}$. Use sum rule with constant multiples.
$f'(x) = 5e^x + \frac{7}{x}$. Use sum rule with constant multiples.
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Evaluate the derivative: $f(x) = e^x$ at $x = \text{ln } 2$.
Evaluate the derivative: $f(x) = e^x$ at $x = \text{ln } 2$.
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$f'(\text{ln } 2) = 2$. $f'(x) = e^x$, so $f'(\text{ln } 2) = e^{\text{ln } 2} = 2$.
$f'(\text{ln } 2) = 2$. $f'(x) = e^x$, so $f'(\text{ln } 2) = e^{\text{ln } 2} = 2$.
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Determine $f'(x)$ if $f(x) = e^x + \text{ln } x$.
Determine $f'(x)$ if $f(x) = e^x + \text{ln } x$.
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$f'(x) = e^x + \frac{1}{x}$. Use sum rule: derivative of each term separately.
$f'(x) = e^x + \frac{1}{x}$. Use sum rule: derivative of each term separately.
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Evaluate the derivative: $f(x) = \text{cos } x$ at $x = \frac{\text{π}}{2}$.
Evaluate the derivative: $f(x) = \text{cos } x$ at $x = \frac{\text{π}}{2}$.
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$f'\big(\frac{\text{π}}{2}\big) = -1$. $f'(x) = -\text{sin } x$, so $f'\big(\frac{\text{π}}{2}\big) = -\text{sin}\big(\frac{\text{π}}{2}\big) = -1$.
$f'\big(\frac{\text{π}}{2}\big) = -1$. $f'(x) = -\text{sin } x$, so $f'\big(\frac{\text{π}}{2}\big) = -\text{sin}\big(\frac{\text{π}}{2}\big) = -1$.
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Evaluate the derivative: $f(x) = \text{ln } x$ at $x = 1$.
Evaluate the derivative: $f(x) = \text{ln } x$ at $x = 1$.
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$f'(1) = 1$. $f'(x) = \frac{1}{x}$, so $f'(1) = \frac{1}{1} = 1$.
$f'(1) = 1$. $f'(x) = \frac{1}{x}$, so $f'(1) = \frac{1}{1} = 1$.
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Evaluate the derivative: $f(x) = \text{ln } x$ at $x = 2$.
Evaluate the derivative: $f(x) = \text{ln } x$ at $x = 2$.
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$f'(2) = \frac{1}{2}$. $f'(x) = \frac{1}{x}$, so $f'(2) = \frac{1}{2}$.
$f'(2) = \frac{1}{2}$. $f'(x) = \frac{1}{x}$, so $f'(2) = \frac{1}{2}$.
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Differentiate $f(x) = e^x$ at $x = 1$.
Differentiate $f(x) = e^x$ at $x = 1$.
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$f'(1) = e$. $f'(x) = e^x$, so $f'(1) = e^1 = e$.
$f'(1) = e$. $f'(x) = e^x$, so $f'(1) = e^1 = e$.
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Differentiate $f(x) = \text{cos } x$ at $x = 0$.
Differentiate $f(x) = \text{cos } x$ at $x = 0$.
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$f'(0) = 0$. $f'(x) = -\text{sin } x$, so $f'(0) = -\text{sin }(0) = 0$.
$f'(0) = 0$. $f'(x) = -\text{sin } x$, so $f'(0) = -\text{sin }(0) = 0$.
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Find the derivative: $y = e^x$.
Find the derivative: $y = e^x$.
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$\frac{dy}{dx} = e^x$. Apply the derivative rule for $e^x$.
$\frac{dy}{dx} = e^x$. Apply the derivative rule for $e^x$.
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What is the derivative of $\text{ln } x$?
What is the derivative of $\text{ln } x$?
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$\frac{1}{x}$. Standard derivative of natural logarithm.
$\frac{1}{x}$. Standard derivative of natural logarithm.
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What is the derivative of $e^x$?
What is the derivative of $e^x$?
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$e^x$. The exponential function is its own derivative.
$e^x$. The exponential function is its own derivative.
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