Change in Tandem - AP Precalculus
Card 1 of 30
Calculate $\frac{d}{dx}(5x^4)$.
Calculate $\frac{d}{dx}(5x^4)$.
Tap to reveal answer
$20x^3$. Constant factor rule: multiply derivative by the constant 5.
$20x^3$. Constant factor rule: multiply derivative by the constant 5.
← Didn't Know|Knew It →
What is the derivative of $\tan^{-1}(x)$?
What is the derivative of $\tan^{-1}(x)$?
Tap to reveal answer
$\frac{1}{1+x^2}$. Standard derivative formula for inverse tangent function.
$\frac{1}{1+x^2}$. Standard derivative formula for inverse tangent function.
← Didn't Know|Knew It →
Calculate $\frac{d}{dx}(\frac{1}{x+2})$.
Calculate $\frac{d}{dx}(\frac{1}{x+2})$.
Tap to reveal answer
$-\frac{1}{(x+2)^2}$. Chain rule: derivative of $u^{-1}$ is $-u^{-2} \cdot u'$.
$-\frac{1}{(x+2)^2}$. Chain rule: derivative of $u^{-1}$ is $-u^{-2} \cdot u'$.
← Didn't Know|Knew It →
Identify the derivative of $\cos^{-1}(x)$.
Identify the derivative of $\cos^{-1}(x)$.
Tap to reveal answer
$-\frac{1}{\sqrt{1-x^2}}$. Standard derivative formula for inverse cosine function.
$-\frac{1}{\sqrt{1-x^2}}$. Standard derivative formula for inverse cosine function.
← Didn't Know|Knew It →
Determine $\frac{d}{dx}(\sin^{-1}(x))$.
Determine $\frac{d}{dx}(\sin^{-1}(x))$.
Tap to reveal answer
$\frac{1}{\sqrt{1-x^2}}$. Standard derivative formula for inverse sine function.
$\frac{1}{\sqrt{1-x^2}}$. Standard derivative formula for inverse sine function.
← Didn't Know|Knew It →
What is the derivative of $\sin^2(x)$ using the Chain Rule?
What is the derivative of $\sin^2(x)$ using the Chain Rule?
Tap to reveal answer
$2\sin(x)\cos(x)$. Chain rule with power rule: $2\sin(x) \cdot \cos(x)$.
$2\sin(x)\cos(x)$. Chain rule with power rule: $2\sin(x) \cdot \cos(x)$.
← Didn't Know|Knew It →
Find the derivative of $x^{\frac{1}{2}}$.
Find the derivative of $x^{\frac{1}{2}}$.
Tap to reveal answer
$\frac{1}{2}x^{-\frac{1}{2}}$. Power rule with fractional exponent: $\frac{1}{2} \cdot x^{-1/2}$.
$\frac{1}{2}x^{-\frac{1}{2}}$. Power rule with fractional exponent: $\frac{1}{2} \cdot x^{-1/2}$.
← Didn't Know|Knew It →
Determine $\frac{d}{dx}(\cot(x))$.
Determine $\frac{d}{dx}(\cot(x))$.
Tap to reveal answer
$-\csc^2(x)$. Standard derivative formula for cotangent function.
$-\csc^2(x)$. Standard derivative formula for cotangent function.
← Didn't Know|Knew It →
Find the derivative of $\csc(x)$.
Find the derivative of $\csc(x)$.
Tap to reveal answer
$-\csc(x)\cot(x)$. Standard derivative formula for cosecant function.
$-\csc(x)\cot(x)$. Standard derivative formula for cosecant function.
← Didn't Know|Knew It →
What is the derivative of $\sec(x)$?
What is the derivative of $\sec(x)$?
Tap to reveal answer
$\sec(x)\tan(x)$. Standard derivative formula for secant function.
$\sec(x)\tan(x)$. Standard derivative formula for secant function.
← Didn't Know|Knew It →
What is the derivative of $\frac{u}{v}$ using the Quotient Rule?
What is the derivative of $\frac{u}{v}$ using the Quotient Rule?
Tap to reveal answer
$\frac{u'v - uv'}{v^2}$. Quotient rule: top derivative times bottom minus top times bottom derivative, over bottom squared.
$\frac{u'v - uv'}{v^2}$. Quotient rule: top derivative times bottom minus top times bottom derivative, over bottom squared.
← Didn't Know|Knew It →
Determine $\frac{d}{dx}(\tan(x))$.
Determine $\frac{d}{dx}(\tan(x))$.
Tap to reveal answer
$\sec^2(x)$. Standard derivative formula for tangent function.
$\sec^2(x)$. Standard derivative formula for tangent function.
← Didn't Know|Knew It →
What is the derivative of $u \cdot v$ using the Product Rule?
What is the derivative of $u \cdot v$ using the Product Rule?
Tap to reveal answer
$u'v + uv'$. Apply product rule to find derivative of product of functions.
$u'v + uv'$. Apply product rule to find derivative of product of functions.
← Didn't Know|Knew It →
State the Product Rule for differentiation.
State the Product Rule for differentiation.
Tap to reveal answer
$(uv)' = u'v + uv'$. Derivative of first times second plus first times derivative of second.
$(uv)' = u'v + uv'$. Derivative of first times second plus first times derivative of second.
← Didn't Know|Knew It →
Identify the derivative of $x^{-1}$.
Identify the derivative of $x^{-1}$.
Tap to reveal answer
$-x^{-2}$. Power rule with negative exponent: $-1 \cdot x^{-2}$.
$-x^{-2}$. Power rule with negative exponent: $-1 \cdot x^{-2}$.
← Didn't Know|Knew It →
Calculate $\frac{d}{dx}(x^5 + x^3)$.
Calculate $\frac{d}{dx}(x^5 + x^3)$.
Tap to reveal answer
$5x^4 + 3x^2$. Apply power rule to each term separately.
$5x^4 + 3x^2$. Apply power rule to each term separately.
← Didn't Know|Knew It →
Find the derivative of $\ln(e^x)$.
Find the derivative of $\ln(e^x)$.
Tap to reveal answer
$1$. Since $\ln(e^x) = x$, the derivative is simply 1.
$1$. Since $\ln(e^x) = x$, the derivative is simply 1.
← Didn't Know|Knew It →
Find the derivative of $\csc(x)$.
Find the derivative of $\csc(x)$.
Tap to reveal answer
$ -\csc(x)\cot(x) $. Standard derivative formula for cosecant function.
$ -\csc(x)\cot(x) $. Standard derivative formula for cosecant function.
← Didn't Know|Knew It →
Calculate $\frac{d}{dx}(5x^4)$.
Calculate $\frac{d}{dx}(5x^4)$.
Tap to reveal answer
$20x^3$. Constant factor rule: multiply derivative by the constant 5.
$20x^3$. Constant factor rule: multiply derivative by the constant 5.
← Didn't Know|Knew It →
What is the formula for $\frac{d}{dx}(\tan(x))$?
What is the formula for $\frac{d}{dx}(\tan(x))$?
Tap to reveal answer
$\sec^2(x)$. Standard derivative formula for tangent function.
$\sec^2(x)$. Standard derivative formula for tangent function.
← Didn't Know|Knew It →
What is the derivative of a constant $c$?
What is the derivative of a constant $c$?
Tap to reveal answer
- Constants have zero rate of change.
- Constants have zero rate of change.
← Didn't Know|Knew It →
Find the derivative of $\ln(e^x)$.
Find the derivative of $\ln(e^x)$.
Tap to reveal answer
$1$. Since $\ln(e^x) = x$, the derivative is simply 1.
$1$. Since $\ln(e^x) = x$, the derivative is simply 1.
← Didn't Know|Knew It →
Find the derivative of $e^{3x}$.
Find the derivative of $e^{3x}$.
Tap to reveal answer
$3e^{3x}$. Chain rule: derivative of $e^u$ is $e^u \cdot u'$, where $u = 3x$.
$3e^{3x}$. Chain rule: derivative of $e^u$ is $e^u \cdot u'$, where $u = 3x$.
← Didn't Know|Knew It →
What does $\frac{dy}{dx}$ represent in relation to change in tandem?
What does $\frac{dy}{dx}$ represent in relation to change in tandem?
Tap to reveal answer
Rate of change of $y$ with respect to $x$. Shows how fast $y$ changes as $x$ changes at any point.
Rate of change of $y$ with respect to $x$. Shows how fast $y$ changes as $x$ changes at any point.
← Didn't Know|Knew It →
State the Chain Rule for differentiation.
State the Chain Rule for differentiation.
Tap to reveal answer
$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Multiplies derivatives when functions are composed inside each other.
$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Multiplies derivatives when functions are composed inside each other.
← Didn't Know|Knew It →
What is the formula for the derivative of $f(g(x))$ using the Chain Rule?
What is the formula for the derivative of $f(g(x))$ using the Chain Rule?
Tap to reveal answer
$f'(g(x)) \cdot g'(x)$. Apply outer function derivative at inner function, then multiply by inner derivative.
$f'(g(x)) \cdot g'(x)$. Apply outer function derivative at inner function, then multiply by inner derivative.
← Didn't Know|Knew It →
Identify the derivative of $\sin(g(x))$ using the Chain Rule.
Identify the derivative of $\sin(g(x))$ using the Chain Rule.
Tap to reveal answer
$\cos(g(x)) \cdot g'(x)$. Derivative of sine is cosine, multiplied by derivative of inner function.
$\cos(g(x)) \cdot g'(x)$. Derivative of sine is cosine, multiplied by derivative of inner function.
← Didn't Know|Knew It →
What is the derivative of $e^x$?
What is the derivative of $e^x$?
Tap to reveal answer
$e^x$. The exponential function is its own derivative.
$e^x$. The exponential function is its own derivative.
← Didn't Know|Knew It →
Determine $\frac{d}{dx}(\ln(x^2))$.
Determine $\frac{d}{dx}(\ln(x^2))$.
Tap to reveal answer
$\frac{2}{x}$. Using chain rule with $\ln(u)$ where $u = x^2$, so $\frac{u'}{u} = \frac{2x}{x^2}$.
$\frac{2}{x}$. Using chain rule with $\ln(u)$ where $u = x^2$, so $\frac{u'}{u} = \frac{2x}{x^2}$.
← Didn't Know|Knew It →
What is the formula for $\frac{d}{dx}(\tan(x))$?
What is the formula for $\frac{d}{dx}(\tan(x))$?
Tap to reveal answer
$\sec^2(x)$. Standard derivative formula for tangent function.
$\sec^2(x)$. Standard derivative formula for tangent function.
← Didn't Know|Knew It →