Definition of the Derivative

Recall from the beginning of Chapter 2, the calculations of the average rate of change and the instantaneous rate of change.   In a graphing interpretation, the average rate of change is the slope of a secant line, and the instantaneous rate of change is the limit of these slopes to find the slope of the tangent line.   More precisely, for a function \(f(x)\) on the interval \([a,x]\):

Average Rate of Change	\(m_{sec}=\frac{f(x)-f(a)}{x-a}\)	slope of secant line
Instantaneous Rate of Change	\(m_{tan}=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}\)	slope of tangent line

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Example

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Alternative Formulas

Instead of looking at values where \(x\rightarrow a\), let \(h\) represent the distance \(x\) is from \(a\), \(x=a+h\).   Then as \(x\) gets close to \(a\), this makes \(h\rightarrow0\).   Now the instantaneous rate of change becomes: $$m_{tan}=\lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{a+h-a}\implies\lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h}$$ Hopefully, this looks familiar to you, it is the Difference Quotient discussed back in Chapter 1!   We can look differently at our example now by considering \(a=1\) and use this alternate formula:

Example

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Derivative as a Function

Consider the following graphs:

The graph on the left is a function with tangent line segments at certain points.   The graph on the right is just the tangent line segments with the function removed.   Whenever a tangent line exists at a point, the \(x\)-value can be associated with the slope of that tangent line.   Since it is a tangent line, that slope is unique for that \(x\)-value, creating a new function called the derivative.   The common notation for the derivative is \(f^\prime(x)\).   Together with limits, we define the derivative of a function \(f(x)\) to be: $$f^\prime(x)=\lim_{h\rightarrow0}\left(\frac{f(x+h)-f(x)}{h}\right)$$ provided the limit exists for each \(x\) in the domain of \(f\).   This is the algebraic method of finding the derivative of a function.

Example

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Derivative Notations

Next to the standard notation for derivative, \(f^\prime(x)\), there are other useful notations that will appear later on: $$\frac{dy}{dx};\;\frac{d}{dx}\left(f(x)\right);\;D_x\left(f(x)\right);\;y^\prime(x)$$ When we need to evaluate the derivative at a value like \(x=a\), then you might see: $$f^\prime(a);\;y^\prime(a);\;\frac{df}{dx}\Bigg\lvert_{x=a};\;\frac{dy}{dx}\Bigg\lvert_{x=a}$$

Example

Find \(f^\prime(1)\) using the function \(f(x)=-16x^2+96x\).

From the preceeding example, \(f^\prime(x)=-32x+96\)	Hence \(f^\prime(1)=-32(1)+96\implies f^\prime(1)=-32+96\implies f^\prime(1)=64\;\color{green}{\checkmark}\)
To see the alternative notation used:	\(\large{\frac{df}{dx}}\normalsize{\Bigg\lvert_{x=1}=64}\)

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