Application of Derivatives
- Continuity and Differentiability of a function: Let a function \[y=f(x)\] is said to be continuous at
\[x=a\] then \[\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f(x)=\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,\,\,f(x)=f(a)\]
Generally, a function is said to be continuous at \[x=a\] when the graph of that function can be drawn/sketched without lefting the pencil.
- Differentiation: The process of finding out the differentiability/derivatives of the function \[y=f(x)\]in the interval (a, b) is said to be differentiation.
- Derivatives of f(x): Let \[y=f(x)\] is continuous in interval [a, b]. Let a point \[c\in (a,b)\]
Then function \[y=f(x)\] is differentiable at \[x=c\]
i.e. \[\underset{x\to c}{\mathop{\lim }}\,\frac{f(x+c)-(c)}{c}=f'(c)\]
Solved Example
Find derivative of \[y=\sin x.\] by 1st principle:
Let \[y=f(x)=\sin x\] ...(1)
Let \[\delta \]x be the small increment in x then \[\delta \]y be the corresponding increment in y.
\[y+\delta y=\sin (x+\delta x)\] ...(2)
Now, on subtracting equation (1) from (2), we get
\[y+\delta y-y=\sin (x+\delta x)-\sin x\]
\[\delta y=2.\cos \frac{x+\delta x+x}{2}.\sin \left( \frac{x+\delta x+x}{2} \right)\]
Dividing \[\delta \]x on both sides and taking limit \[\delta x\to 0,\]we get
\[\underset{\delta x\to 0}{\mathop{\lim }}\,\frac{\delta y}{\delta x}=\underset{\delta x\to 0}{\mathop{\lim }}\,2.\frac{\cos \left( \frac{2x+\delta x}{2} \right).\sin \left( \frac{\delta x}{2} \right)}{\delta x}\]
\[\frac{dy}{dx}\underset{\delta x\to 0}{\mathop{\lim }}\,2.\cos \left( x+\frac{\delta x}{2} \right).\left( \frac{\sin \frac{\delta x}{2}}{\frac{\delta x}{2}\times 2} \right)\]
On applying limit \[\delta x\to 0,\]we get
\[\frac{dy}{dx}=\cos x\times 1=\cos x\left[ \underset{\theta \to 0}{\mathop{\lim }}\,\frac{\sin \theta }{\theta }=1 \right]\]
Thus,\[\frac{d}{dx}(\sin x)=\cos x\]
\[\frac{dy}{dx}\]is said to be differential coefficient of \[y=f(x).\] It is denoted by\[{{y}_{1}}\]or f \['(x)\]
\[\frac{d}{dx}(\sin x)=\cos x\]
\[\frac{d}{dx}(\cos x)=-\sin x\]
\[\frac{d}{dx}(\tan x)={{\sec }^{2}}x\]
\[\frac{d}{dx}(\cot x)=-\cos e{{c}^{2}}x\]
\[\frac{d}{dx}(\sec x)=\sec x\tan x\]
\[\frac{d}{dx}(\cos ec\,x)=-\cos ec\,x.\cot x\]
\[\frac{d}{dx}({{x}^{n}})=n{{x}^{n-1}}\]
\[\frac{d}{dx}({{e}^{x}})={{e}^{x}}\]
\[\frac{d}{dx}(\log x)=\frac{1}{x}\]
\[\frac{d}{dx}(si{{n}^{-1}}x)=\frac{1}{\sqrt{1-{{x}^{2}}}}\]
\[\frac{d}{dx}(co{{s}^{-1}}x)=\frac{-1}{\sqrt{1-{{x}^{2}}}}\]
\[\frac{d}{dx}(ta{{n}^{-1}}x)=\frac{1}{1+{{x}^{2}}}\]
\[\frac{d}{dx}(x)=1\]
\[\frac{d}{dx}(C)=0\] where \[C=\]any const.
\[\frac{d}{dx}(\sin ax)=a\,\text{cos}\,ax\]
\[\frac{d}{dx}({{a}^{x}})={{a}^{x}}.\log a\]
- Some Basic Rules of Differentiation
\[\frac{d}{dx}(u\,\pm v)=\frac{d}{dx}(u)\pm \frac{d}{dx}(v)\]where u and v be the function of x.
\[\frac{d}{dx}(C.u)=C.\frac{d}{dx}(u)\] where \[C=\]any const.
e.g. \[\frac{d}{dx}(5{{x}^{2}})=5.\frac{d}{dx}({{x}^{2}})=5[2.{{(x)}^{2-1}}]=10{{x}^{1}}=10x\]
\[\frac{d}{dx}(u.v.)=u.\frac{d\text{v}}{dx}+\frac{du}{dx}\]
e.g. \[\frac{d}{dx}({{e}^{x}}.\sin x)={{e}^{x}}.\frac{d}{dx}(\sin x)+\sin x\frac{d}{dx}({{e}^{x}})={{e}^{x}}\cos x+\sin x.{{e}^{x}}\]
\[={{e}^{x}}(\cos x+\sin x)\]
\[\frac{d}{dx}\left( \frac{u}{\text{v}} \right)=\frac{\text{v}.\frac{du}{dx}-u.\frac{d\text{v}}{dx}}{{{\text{v}}^{2}}}\] e.g. \[\frac{d}{dx}\left( \frac{{{x}^{2}}}{\sin x} \right)_{\text{v}}^{u}=\frac{\sin x\frac{d}{dx}({{x}^{2}})-{{x}^{2}}\frac{d}{dx}(\sin x)}{{{\sin }^{2}}x}\]
\[\frac{d}{dx}(lo{{g}_{c}}{{a}^{x}})=\frac{d}{dx({{a}^{x}})}(\log {{a}^{x}}).\frac{d}{dx}({{\underline{a}}^{x}})=\frac{1}{{{a}^{x}}}.{{a}^{x}}.\log a=\log a\]
Geometrical meaning of derivative at point: The derivative of a function \[f(x)\] at a point \[x=a\] is the of the tangent of the curve \[y=f(x)\] at the point \[(a,f(a)).\]
Let us consider a curve \[y=f(x)\] & take a point P\[(x,y)\] on it. We draw the tangent to the curve at \[P(x,y)\]which makes an angle \[\alpha \] with positive direction of x-axis. Then,
\[{{\left. \frac{dy}{dx} \right|}_{\,\,at\,\,P(x,y)\,=\,tan\,\,\alpha \,=\,m(say)}}\]
It is said to be gradient or slope of the tangent to the curve \[y=f(x)\]at\[p(x,y)\].
- Equation of the tangent: The equation of the tangent to a curve \[y=f(x)\] at the given point \[P({{x}_{1}},{{y}_{1}})\] is written in point slope form of the equation of straight more...