Differentiation - Practice Questions & MCQ

DIFFERENTIATION

The rate of change of a quantity y with respect to another quantity x is called the derivative or differential coefficient of y with respect to x. Geometrically, the Differentiation of a function at a point represents the slope of the tangent to the graph of the function at that point.

Consider any function, $y=f(x)$
Let $\mathrm{P}(\mathrm{x}, \mathrm{y})$ and $\mathrm{Q}(\mathrm{x}+\Delta \mathrm{x}, \mathrm{f}(\mathrm{x}+\Delta \mathrm{x}))$ be two points on the curve

Slope of chord PQ $=\frac{f(x+\Delta x)-f(x)}{(x+\Delta x)-x}=\frac{f(x+\Delta x)-f(x)}{\Delta x}$
(Using two points lying on the straight line PQ)

Now as $\Delta x \rightarrow 0$, the point $Q$ moves infinitesimally close to point $P$, and chord $P Q$ becomes tangent to the curve at point $P$.

So slope of chord PQ becomes the slope of the tangent to the curve at point $P$
So slope of tangent at $\mathrm{P}=\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}$
This gives the slope of the tangent at P. This value is also known as the derivative or differentiation of function $f(x)$ concerning $x$.

It is also denoted by $\frac{d \mathbf{f}(\mathbf{x})}{d \mathbf{x}}$ or $\frac{\mathrm{dy}}{\mathrm{dx}}$ or $\mathbf{f}^{\prime}(\mathbf{x})$ or $D(\mathbf{f}(\mathbf{x}))$
If we replace $\Delta x$ with $h$, then we can write

$
\begin{aligned}
\frac{\mathrm{dy}}{\mathrm{dx}}=\lim _{\Delta \mathrm{x} \rightarrow 0} & \frac{\mathrm{f}(\mathrm{x}+\Delta \mathrm{x})-\mathrm{f}(\mathrm{x})}{\Delta \mathrm{x}} \\
& =\lim _{h \rightarrow 0} \frac{\mathrm{f}(\mathrm{x}+h)-\mathrm{f}(\mathrm{x})}{h}
\end{aligned}
$
This method to find the derivative of $f(x)$ is also known as the "First Principle of derivative" or "a-b initio method"