9f0005…: “Next to the interesting owl”

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Worked Out Examples Problem Videos

The following video may be helpful when trying to solve this example. Note that you may skip to the end of the video to get completion credit for this page if you don’t need to watch it.

Practice Problems: Derivatives as Functions

1 : Compute the derivative of $f(x)=\sage {p1f1}$ using the definition of the derivative, leaving it in terms of a general function (i.e. it should be a function rather than a value).

The derivative of $f(x)$ is: $\answer {\sage {p1ans}}$ .

Remember you want to use the definition of a derivative but leave the functions in terms of $x$ , so you want to solve the difference quotient: $\lim \limits _{\Delta x \rightarrow 0}\frac {f\left (x + \Delta x\right ) - f\left (x\right )}{\Delta x}$ It may help to not actually simplify much for $f\left (x + \Delta x\right )$ and $f\left (x\right )$ terms individually, as there will be a lot of cancellation between them, so you may want to write it out long-way and see how much canceling you can do term-wise before you try to simplify! Remember, when in doubt, expand everything as much as you can and combine like terms, or if you have a radical, you may need to rationalize the numerator.