| Differentials | Error Propagation |
| Linear Approximation | More Practice |
Differentials, Linear Approximation, and Error Propagation are more applications of Differential Calculus.
Differentials
Think of differentials of picking apart the “fraction” $ \displaystyle \frac{{dy}}{{dx}}$ we learned to use when differentiating a function.
We learned that the derivative or rate of change of a function can be written as $ \displaystyle \frac{{dy}}{{dx}}={f}’\left( x \right)$, where $ dy$ is an infinitely small change in $ y$, and $ dx$ (or $ \Delta x$) is an infinitely small change in $ x$. It turns out that if $ f\left( x \right)$ is a function that is differentiable on an open interval containing $ x$, and the differential of $ x$ ($ dx$) is a non-zero real number, then $ dy={f}’\left( x \right)dx$ (see how we just multiplied both sides by $ dx$)? And I won’t get into this at this point, but the differential of $ y$ can be used to approximate the change in $ y$, so $ \Delta y\approx dy$. (I always forget this, but remember that $ \Delta y=f\left( {x+\Delta x} \right)-f\left( x \right)$.) Anyway, all this can get really theoretical really fast, so I’m leaving out a lot, but hopefully this will be enough to work the problems.
Calculating Differentials
We learned differentiation rules earlier, and these apply to differentials too. These look familiar, right? We’ll see that we’ll need to use the differential produce rule in the problem here. Here are the differential formulas:
If $ u$ and $ v$ are differentiable functions of $ x$:
Constant Multiple: $ d\left[ {cu} \right]=c\,du$
Sum/Difference: $ d\left[ {u\pm v} \right]=du\pm dv$
Product: $ d\left[ {uv} \right]=u\,dv+v\,du$
Quotient: $ \displaystyle d\left[ {\frac{u}{v}} \right]=\frac{{v\,du-udv}}{{{{v}^{2}}}}$
Linear Approximation
We can use differentials to perform linear approximations of functions, like we did with tangent lines here in the Equation of the Tangent Line, Tangent Line Approximation, and Rates of Change section. This formula that looks similar to a point-slope formula, since a derivative is a slope: $ y-{{y}_{0}}={f}’\left( {{{x}_{0}}} \right)\left( {x-{{x}_{0}}} \right)$, or $ f\left( x \right)-f\left( {{{x}_{0}}} \right)={f}’\left( {{{x}_{0}}} \right)\left( {x-{{x}_{0}}} \right)$, which means $ f\left( x \right)=f\left( {{{x}_{0}}} \right)+{f}’\left( {{{x}_{0}}} \right)\left( {x-{{x}_{0}}} \right)$. (Remember that the variables with subscript “0” are the “old” or “original” values.)
Think of this equation as the “new $ y$” equals the “old $ y$” plus the derivative at the “old $ x$” times the difference between the “new $ x$” and the “old $ x$”.
Note: Another way we can think of differentials is using this formula; some teachers prefer this way: $ \displaystyle \frac{{dy}}{{dx}}={f}’\left( x \right);\,\,dy={f}’\left( x \right)dx$ (this makes sense, right? A “slope” is a “slope”). Then once we get $ dy$, we just add it back to the original $ y$ to get the approximation. This is also shown in the fourth problem below.
Here are some examples in both finding differentials and finding approximations of functions:
Error Propagation
We can also use differentials to estimate errors, for example, for error analyses in Physics. In these problems, we’ll typically take a derivative, and use the “$ dx$” or “$ dy$” part of the derivative as the error. Then, to get percent error, we’ll divide the error by the total amount and multiply by 100. When solving for error, it can go either way, so we typically express our answers with a “$ \pm $”.
Attack these problems the same way we did with Related Rates problems: write down what we know, what we need, and how we relate the variables. Typically, the words “is correct within” signifies error. Here are some problems; you’ll have to use some Geometry:
Here are a few more that are a bit more difficult; for the first below, we need to use the Differential Product Rule:
Divide this by the total area to get the percent area: .
Learn these rules and practice, practice, practice!
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On to Exponential and Logarithmic Differentiation — you are ready!