Differentiability - Notes, Concept and All Important Formula

DIFFERENTIABILITY

1. INTRODUCTION:

The derivative of a function 'f' is function ; this function is denoted by symbols such as

$\mathrm{f}^{\prime}(\mathrm{x}), \dfrac{\mathrm{df}}{\mathrm{dx}}, \dfrac{\mathrm{d}}{\mathrm{dx}} \mathrm{f}(\mathrm{x})$ or $\dfrac{\mathrm{df}(\mathrm{x})}{\mathrm{dx}}$

The derivative evaluated at a point a, can be written as:

$\mathrm{f}^{\prime}(\mathrm{a}),\left[\dfrac{\mathrm{df}(\mathrm{x})}{\mathrm{dx}}\right]_{\mathrm{x}=\mathrm{a}}, \mathrm{f}^{\prime}(\mathrm{x})_{\mathrm{x}=\mathrm{a}}$, etc.

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All Chapter Notes, Concept and Important Formula

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2. RIGHT HAND & LEFT HAND DERIVATIVES:

(a) Right hand derivative :

The right hand derivative of $\mathrm{f}(\mathrm{x})$ at $\mathrm{x}=$ a denoted by $\mathrm{f}_{+}^{\prime}(\mathrm{a})$ is defined as :

$\mathrm{f}_{+}^{\prime}(\mathrm{a})=\displaystyle \lim_{\mathrm{h} \rightarrow 0^{+}} \dfrac{\mathrm{f}(\mathrm{a}+\mathrm{h})-\mathrm{f}(\mathrm{a})}{\mathrm{h}}$, provided the limit exists & is finite.

(b) Left hand derivative :

The left hand derivative of $f(x)$ at $x=$ a denoted by $f^{\prime}(a)$ is defined as $: f_{-}^{\prime}(a)=\displaystyle \lim_{h \rightarrow 0^{+}} \dfrac{f(a-h)-f(a)}{-h}$, provided the limit exists & is finite.

(c) Derivability of function at a point :

If $f_{+}^{\prime}(a)=f^{\prime}(a)=$ finite quantity, then $f(x)$ is said to be derivable or differentiable at $\mathbf{x}=\mathbf{a}$. In such case $\mathrm{f}_{+}^{\prime}(\mathrm{a})=\mathrm{f}_{-}^{\prime}(\mathrm{a})=\mathrm{f}^{\prime}(\mathrm{a})$ & it is called derivative or differential coefficient of $\mathrm{f}(\mathrm{x})$ at $\mathrm{x}=\mathrm{a}$.

Note:

(i) All polynomial, trigonometric, inverse trigonometric, logarithmic and exponential function are continuous and differentiable in their domains, except at end points.

(ii) If $f(x)$ & $g(x)$ are derivable at $x=$ a then the functions $\mathrm{f}(\mathrm{x})+\mathrm{g}(\mathrm{x}), \mathrm{f}(\mathrm{x})-\mathrm{g}(\mathrm{x}), \mathrm{f}(\mathrm{x}) \cdot \mathrm{g}(\mathrm{x})$ will also be derivable at $\mathrm{x}=\mathrm{a}$ & if $g(a) \neq 0$ then the function $f(x) / g(x)$ will also be derivable at $x=a$.

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3. IMPORTANT NOTE :

(a) Let $\mathrm{f}_{+}^{\prime}(\mathrm{a})=\mathrm{p}$ & $\mathrm{f}^{\prime}(\mathrm{a})=\mathrm{q}$

When p & q are finite:

If $\mathrm{p}$ and $\mathrm{q}$ are finite (whether equal or not), then $f$ is continuous at $\mathrm{x}=\mathrm{a}$ but converse is NOT necessarily true.

(i) $\mathrm{p}=\mathrm{q} \Rightarrow \mathrm{f}$ is differentiable at $\mathrm{x}=\mathrm{a} \Rightarrow \mathrm{f}$ is continuous at $\mathrm{x}=\mathrm{a}$ $f^{\prime}(0)=0$ and $f_{+}^{\prime}(0)=0 \Rightarrow \quad f^{\prime}(0)=0$, here $\mathrm{x}$ axis is tangent to the curve at $\mathrm{x}=0$.

(ii) $\mathrm{p} \neq \mathrm{q} \Rightarrow \mathrm{f}$ is not differentiable at $\mathrm{x}=\mathrm{a}$, but $\mathrm{f}$ is still continuous at $\mathrm{x}=\mathrm{a}$. In this case we have a sudden change in the direction of the graph of the function at $\mathrm{x}=\mathrm{a}$. This point is called a corner point of the function. At this point there is no tangent to the curve.

When p or q may not be finite :

In this case $\mathrm{f}$ is not differentiable at $\mathrm{x}=\mathrm{a}$ and nothing can be concluded about continuity of the function at $\mathrm{x}=\mathrm{a}$.

Note:

$\circ$ Corner : If $\mathrm{f}$ is continuous at $\mathrm{x}=$ a with $\mathrm{RHD}$ and $\mathrm{LHD}$ at $\mathrm{x}=\mathrm{a}$ both are finite but not equal or exactly one of them is infinite, then the point $\mathrm{x}=\mathrm{a}$ is called a corner point and at this point function is not differentiable but continuous.

$\circ$ Cusp : If $\mathrm{f}$ is continuous at $\mathrm{x}=\mathrm{a}$ and one of $\mathrm{RHD}, \mathrm{LHD}$ at $x=a$, approaches to $\infty$ and other one approaches to $-\infty$, then the point $\mathrm{x}=\mathrm{a}$ is called a cusp point. At cusp point we have a vertical tangent and at this point function is not differentiable but continuous. We can observe that cusp is sharper than corner point.

(b) Geometrical interpretation of differentiability:

(i) If the function $y=f(x)$ is differentiable at $x=a$, then a unique tangent can be drawn to the curve $y=f(x)$ at $P(a,$, $\mathrm{f}(\mathrm{a}))$ & $\mathrm{f}^{\prime}(\mathrm{a})$ represent the slope of the tangent at point $\mathrm{P}$.

(ii) If LHD and RHD are finite but unequal then it geometrically implies a sharp corner at $\mathrm{x}=\mathrm{a}$. e.g. $f(x)=|x|$ is continuous but not differentiable at $\mathrm{x}=0$.

A sharp corner is seen at $\mathrm{x}=0$ in the graph of $f(x)=|x|$.

(c) Vertical tangent : If $y=f(x)$ is continuous at $x=$ a and $\displaystyle \lim _{x \rightarrow a}\left|f^{\prime}(x)\right|$ approaches to $\infty$, then $y=f(x)$ has a vertical tangent at $x=a$. If a function has vertical tangent at $\mathrm{x}=\mathrm{a}$ then it is non differentiable at $x=a$.

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4. DERIVABILITY OVER AN INTERVAL :

(a) $\mathrm{f}(\mathrm{x})$ is said to be derivable over an open interval $(\mathrm{a}, \mathrm{b})$ if it is derivable at each & every point of the open interval $(\mathrm{a}, \mathrm{b})$

(b) $\mathrm{f}(\mathrm{x})$ is said to be derivable over the closed interval $[\mathrm{a}, \mathrm{b}]$ if :

(i) $\mathrm{f}(\mathrm{x})$ is derivable in $(\mathrm{a}, \mathrm{b})$ & 
(ii) for the points a and $b, f^{\prime}$ $\left(\right.$ a) &  $\mathrm{f}^{\prime}$ (b) exist.

Note:

(i) If $\mathrm{f}(\mathrm{x})$ is differentiable at $\mathrm{x}=\mathrm{a}$ & $\mathrm{~g}(\mathrm{x})$ is not differentiable $\mathrm{at}$ $x=a$, then the product function $F(x)=f(x) \cdot g(x)$ can still be differentiable at $\mathrm{x}=\mathrm{a}$.

(ii) If $f(x)$ & $g(x)$ both are not differentiable at $x=a$ then the product function; $\mathrm{F}(\mathrm{x})=\mathrm{f}(\mathrm{x}) \cdot \mathrm{g}(\mathrm{x})$ can still be differentiable at $\mathrm{x}=\mathrm{a}$.

(iii) If $f(x)$ & $g(x)$ both are non-derivable at $x=a$ then the sum function $\mathrm{F}(\mathrm{x})=\mathrm{f}(\mathrm{x})+\mathrm{g}(\mathrm{x})$ may be a differentiable function.

(iv) If $\mathrm{f}(\mathrm{x})$ is derivable at $\mathrm{x}=\mathrm{a} \Rightarrow \mathrm{f}^{\prime}(\mathrm{x})$ is continuous at $\mathrm{x}=\mathrm{a}$.

(v) Sum or difference of a differentiable and a non-differentiable function is always is non-differentiable.

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