Concept: The derivative is a ratio of small changes

Prerequisite concepts

Understanding both difference (how far apart two values are at one time) and change (how far apart the value of a single parameter is at two different times) is necessary for understanding derivatives.

The derivative relates how much $f$ changes as $x$ changes The derivative is a ratio of small changes The derivative can be approximated by the slope of a secant line The derivative is a limit The derivative is a function The derivative of a constant is zero The derivative is a linear operator Power law The derivative at a cusp is undefined Variables can be held constant Product rule Single variable chain rule You can flip a derivative "With respect to what" matters The magnitude of the derivative at a point is the slope of a tangent line at that point

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The derivative is a ratio of small changes

One can view $\frac{df}{dx}$ as approximately given by a fraction where $df$ is a small change in $f$ and $dx$ is a small change in $x$.

Representations used

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$\frac{\Delta f}{\Delta x}$

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The derivative is a limit

Technically, the derivative is a ratio of small changes in the limit that the change in the denominator goes to zero: $\frac{df}{dx}=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$

You can flip a derivative

In other words, $\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}$. Note that this concept is challenging to express in Newton's notation, but arises naturally if implicit differentiation is covered.

"With respect to what" matters

Taking derivatives of the same function with respect to different variables will produce different results. In physics, these different results might have different dimensions and thus the derivative can have wildly different physical interpretations based off of "'with respect to' what".

Difference / Change The derivative relates how much $f$ changes as $x$ changes The derivative is a ratio of small changes The derivative can be approximated by the slope of a secant line The derivative is a function The derivative of a constant is zero The derivative is a linear operator Power law The derivative at a cusp is undefined Variables can be held constant Product rule Single variable chain rule The magnitude of the derivative at a point is the slope of a tangent line at that point

Partial Derivative Machine Derivatives

There are experimental limits to how $small$ of a change can be measured

While it is relatively easy to imagine very, very small changes in physical values, there are often experimental limits on how small of a change can be measured. When designing and conducting experiments, there is a tension between these experimental limitations and normative representations of functions as smooth lines or

The coefficients in a differentials equation are partial derivatives

FIXME

The derivative can be interpreted physically Partial derivatives that do not have the same variable(s) held constant (at the same values?) are not the same derivative

The Heater II

Derivatives can be found while holding one or more variables constant There is a partial derivative in every direction at any point

Visualizing Divergence

The divergence is related to the total flux through a closed surface The divergence is a function The divergence is a local quantity

Visualizing Curl

Zapping with d

Differentials are small changes or differences Equations can be 'zapped with d' to relate differentials Individual differentials can be manipulated algebraically Total differentials are linear

Chain Rules

There might be experimental limits on which quantities you can measure Differentials allow the finding of partial derivatives when a variable cannot be solved for You can flip a partial derivative if the same variable(s) are constant