Origin of Differential Calculus

Differential calculus is a theory originated from the solution of two old problems:

  1. drawing a tangent line to a curve
  2. calculating the velocity of non-uniform motion of a particle

In both the problems, continuous curves are involved and the limiting process is used. So the objects of study in the differential calculus are continuos functions. These problems were solved in a certain sense by Sir Isaac Newton and Gottfried Wilhelm Leibniz and in the process, differential calculus was discovered.

Tangent Line to a Curve

Let $AB$ be a continuos curve given by $y=ƒ(x)$ and let $P$ and $Q$ be any two points in the curve. Let the co-ordinates of $P$ and $Q$ be $(x, y)$ and $(x’, y’)$.

History of differential calculus: drawing a tangent line to a curve

When a point moves along the curve from the point $P$ to the point $Q$, it moves horizontally through the distance $PR$ and vertically through the distance $RQ$. \[PR=LM=OM-OL=x’-x\] \[RQ=QM-RM=y’-PL=y’-y\]

These quantities $x’-x$ and $y’-y$ are called the increments in $x$ and $y$ respectively and are denoted by $\Delta x$ and $\Delta y$ i.e. $\Delta x=x’-x$ and $\Delta y=y’-y$.

Also, \[\Delta y=ƒ(x’)-ƒ(x)=ƒ(x+\Delta x)-ƒ(x)\]

If we join the points $P$ and $Q$, we get secant $PQ$ which makes an angle $\theta$ with the $\text{X-axis}$, i.e. $\angle QNM=\theta$. So $\angle QPR=\angle QNM=\theta$ and

\[\tan\theta=\frac{QR}{PR}=\frac{\Delta y}{\Delta x}\]

which is the slope of the secant $PQ$. As $Q$ moves along the curve and approaches $P$, the secant rotates about $P$. The limiting position of the secant, when $Q$ ultimately coincides with $P$, is the tangent at $P$, making the angle $\phi$ with the $\text{X-axis}$. In that situation, $\Delta x$, $\Delta y$ tend to zero. So

\[\lim_{\Delta x \to 0} \frac{\Delta x}{\Delta y}=\lim_{\Delta x \to 0} \tan\theta=\tan\phi\]

\[\text{or, } \lim_{\Delta x \to 0}\frac{\Delta y}{\Delta x}=\lim_{\Delta x \to 0} \frac{ƒ(x+\Delta x)-ƒ(x)}{\Delta x}=\tan\phi\]

Therefore, $\lim_{\Delta x\to 0} \frac{\Delta y}{\Delta x}$ or $\lim_{\Delta x\to 0} \frac{ƒ(x+\Delta x)-ƒ(x)}{\Delta x}$ gives the slope of a tangent to the curve given by the function $ƒ$.

Instantaneous Velocity

Suppose a particle is moving in a straight line. Then the distance covered by the particle increases with time. So the distance $s$ can be considered to be a function $ƒ$ of the time $t$, and $s=ƒ(t)$.

History of differential calculus: calculating the velocity of non-uniform motion of a particle

At times $t$ and $t+\Delta t$, suppose the particle is at the points $P$ and $Q$ respectively such that $AP=s$ and $AQ=s+\Delta s$. Then, \[PQ=AQ-AP=s+\Delta s-s=\Delta s\]

Also, \[\Delta s=s+\Delta s-s=ƒ(t+\Delta t)-ƒ(t)\]

So the average velocity, $v_{av}$, during the time interval $(t, t+\Delta t)$ is

\[v_{av}=\frac{\Delta s}{\Delta t}=\frac{ƒ(t+\Delta t)-ƒ(t)}{\Delta t}\]

Now as $\Delta t \to 0$, $Q$ tends to $P$. So the instantaneous velocity $v$ of the particle at $P$ or in time $t$ is the limit to which $v_{av}$ tends as $\Delta t \to 0$, and

\[v=\lim_{\Delta t \to 0}\frac{\Delta s}{\Delta t}=\lim_{\Delta t \to 0} \frac{ƒ(t+\Delta t)-ƒ(t)}{\Delta t}\]

Learn more about Average and Instantaneous Velocity : Motion in a Straight Line (Kinematics)


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