Defining the Derivative
Defining the Derivative
Summary
The slope \(m\) of a line given two points \((x_1, y_1)\) and \((x_2,y_2)\) is given by
\begin{equation*} m=\frac{y_2-y_1}{x_2-x_1}\text{.} \end{equation*}Now we wish to study how to compute the slope of a general curve \(y = f(x)\) at a given point \(x=a\). This is where the concept of secant lines comes in. A secant line is formed by connecting any two points \((a,f(a))\)and \((a+h,f(a+h))\) that lie on the curve\(y = f(x)\), and the slope of the secant line is calculated as above.
Since we know how to compute slopes of lines we use this information to compute the slope of a general curve. Thus as \(h\rightarrow0\) secant lines are created that contain the points \((a,f(a))\) and \((a+h,f(a+h))\). These secant lines have slopes
\begin{equation*} m = \frac{f(a+h)-f(a)}{a+h-a}=\frac{f(a+h)-f(a)}{h}\text{.} \end{equation*}
Thus as \(h\rightarrow0\) the slopes of the secant lines are approaching the slope of the tangent line that contains the point \(x=a\) or \((a,f(a))\). The slope of the tangent line at point \(x=a\) or \((a,f(a))\) is denoted as \(f'(a)\) and is calculated as follows:
\begin{equation*} f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h} \end{equation*}
See the Desmos demonstration on Secant and Tangent Lines.
The slope of the tangent line to a curve measures the instantaneous rate of change of a curve. The derivative, \(f'(a)\) of a function \(f(x)\) at a value a is found using the definition for the slope of the tangent line. Velocity is the rate of change of position. As such, the velocity \(v(t)\) at time \(t\) is the derivative of the position \(s(t)\) at time \(t\). The average velocity over a given interval \([a,b]\) is given by
\begin{equation*} \mathrm{avg}_\mathrm{vel}=\frac{s(b)-s(a)}{b-a} \end{equation*}
which can be interpreted as the slope of the secant line that contains the points \((a,s(a)\) and \((b,s(b)\). Therefore, the instantaneous velocity at time \(t=a\) is given by
\begin{equation*} v(a)=s'(a)=\lim_{h\to 0}\frac{s(a+h)-s(a)}{h} \end{equation*}
We may estimate a derivative by using a table of values.