Differentiation flashcards

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What is the Product Rule for differentiation?

If $y = uv$, where $u$ and $v$ are functions of $x$, then $\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}$.

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What is the Quotient Rule for differentiation?

If $y = \frac{u}{v}$, where $u$ and $v$ are functions of $x$, then $\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$.

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How do you differentiate $f(y)$ with respect to $x$?

Using the chain rule: $\frac{d}{dx}(f(y)) = f'(y) \times \frac{dy}{dx}$. This is the core principle of implicit differentiation.

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What is the formula for the gradient of a parametric curve?

For a curve defined by $x = f(t)$ and $y = g(t)$, the gradient is $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$.

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What is the derivative of $\ln(x)$?

$\frac{d}{dx}(\ln x) = \frac{1}{x}$

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What is the derivative of $a^x$?

$\frac{d}{dx}(a^x) = a^x \ln a$. A special case is $\frac{d}{dx}(e^x) = e^x$ since $\ln e = 1$.

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What is the derivative of $\tan(x)$?

$\frac{d}{dx}(\tan x) = \sec^2 x$

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What is the derivative of $\arcsin(x)$?

$\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}$

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What is the derivative of $\arctan(x)$?

$\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}$

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Common trap: What is a frequent mistake in implicit differentiation?

Forgetting to multiply by $\frac{dy}{dx}$ after differentiating a term involving $y$. For example, the derivative of $y^2$ with respect to $x$ is $2y \frac{dy}{dx}$, not just $2y$.

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What is the first step in a related rates problem?

Identify the variables involved and find a static equation that connects them, before differentiating with respect to time, $t$.