Skip to main content

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.


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 \(x=a\).


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.

Vertical tangent concept

(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 \(\mathrm{P}\).

Geometrical interpretation of differentiability

(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 \(x=a\).


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.

Labels:

Comments

Post a Comment

Popular posts from this blog

Indefinite Integration - Notes, Concept and All Important Formula

INDEFINITE INTEGRATION If  f & F are function of \(x\) such that \(F^{\prime}(x)\) \(=f(x)\) then the function \(F\) is called a PRIMITIVE OR ANTIDERIVATIVE OR INTEGRAL of \(\mathrm{f}(\mathrm{x})\) w.r.t. \(\mathrm{x}\) and is written symbolically as \(\displaystyle \int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}\) \(=\mathrm{F}(\mathrm{x})+\mathrm{c} \Leftrightarrow \dfrac{\mathrm{d}}{\mathrm{dx}}\{\mathrm{F}(\mathrm{x})+\mathrm{c}\}\) \(=\mathrm{f}(\mathrm{x})\) , where \(\mathrm{c}\) is called the constant of integration. Note : If \(\displaystyle \int f(x) d x\) \(=F(x)+c\) , then \(\displaystyle \int f(a x+b) d x\) \(=\dfrac{F(a x+b)}{a}+c, a \neq 0\) All Chapter Notes, Concept and Important Formula 1. STANDARD RESULTS : (i) \( \displaystyle \int(a x+b)^{n} d x\) \(=\dfrac{(a x+b)^{n+1}}{a(n+1)}+c ; n \neq-1\) (ii) \(\displaystyle \int \dfrac{d x}{a x+b}\) \(=\dfrac{1}{a} \ln|a x+b|+c\) (iii) \(\displaystyle \int e^{\mathrm{ax}+b} \mathrm{dx}\) \(=\dfrac{1}{...
Post a Comment
Read more

Logarithm - Notes, Concept and All Important Formula

LOGARITHM LOGARITHM OF A NUMBER : The logarithm of the number \(\mathrm{N}\) to the base ' \(\mathrm{a}\) ' is the exponent indicating the power to which the base 'a' must be raised to obtain the number \(\mathrm{N}\) . This number is designated as \(\log _{\mathrm{a}} \mathrm{N}\) . (a) \(\log _{a} \mathrm{~N}=\mathrm{x}\) , read as \(\log\) of \(\mathrm{N}\) to the base \(\mathrm{a} \Leftrightarrow \mathrm{a}^{\mathrm{x}}=\mathrm{N}\) . If \(a=10\) then we write \(\log N\) or \(\log _{10} \mathrm{~N}\) and if \(\mathrm{a}=e\) we write \(\ln N\) or \(\log _{e} \mathrm{~N}\) (Natural log) (b) Necessary conditions : \(\mathrm{N}> \,\,0 ; \,\, \mathrm{a}> \,\,0 ; \,\, \mathrm{a} \neq 1\) (c) \(\log _{a} 1=0\) (d) \(\log _{a} a=1\) (e) \(\log _{1 / a} a=-1\) (f) \(\log _{a}(x . y)=\log _{a} x+\log _{a} y ; \,\, x, y> \,\,0\) (g) \(\log _{a}\left(\dfrac{\mathrm{x}}{y}\right)=\log _{\mathrm{a}} \mathrm{x}-\log _{\mathrm{a}} \mathrm{y} ; \,\, \mathrm{...
Post a Comment
Read more

Trigonometry Equation - Notes, Concept and All Important Formula

TRIGONOMETRIC EQUATION 1. TRIGONOMETRIC EQUATION : An equation involving one or more trigonometrical ratios of unknown angles is called a trigonometric equation. All Chapter Notes, Concept and Important Formula 2. SOLUTION OF TRIGONOMETRIC EQUATION : A value of the unknown angle which satisfies the given equations is called a solution of the trigonometric equation. (a) Principal solution :- The solution of the trigonometric equation lying in the interval \([0,2 \pi]\) . (b) General solution :- Since all the trigonometric functions are many one & periodic, hence there are infinite values of \(\theta\) for which trigonometric functions have the same value. All such possible values of \(\theta\) for which the given trigonometric function is satisfied is given by a general formula. Such a general formula is called general solutions of trigonometric equation. 3. GENERAL SOLUTIONS OF SOME TRIGONOMETRICE EQUATIONS (TO BE REMEMBERED) :   (a) If \(\sin \theta=0\) , then \(\theta=...
Post a Comment
Read more