Differentiation¶
Notations used for Derivative¶
Name | Notation Used |
---|---|
Leibniz | $$ \frac{dy}{dx} \text{ or } \frac{df}{dx} $$ |
Newton | $$ \dot{f}(x) $$ |
Lagrange | $$ f'(x) $$ |
Cauchy | $$ D f(x) $$ |
Definition of Derivative¶
If \(y = f(x)\), the derivative of \(y\) with respect to \(x\) is defined as:
- \(\frac{dy}{dx}=\lim\limits_{h\to0}\frac{f(x+h)-f(x)}{h}\)
- \(\frac{d}{dx}f(x)=\lim\limits_{\delta x\to0}\frac{f(x+\delta x)-f(x)}{\delta x}\)
- \(f'(x)=\lim\limits_{x\to a}\frac{f(x)-f(x)}{x-a}\)
The process of taking derivative is called differentiation.
General Rules of Differentiation¶
- \(\frac{d}{dx}(c)=0\), where \(c\) is a constant.
- \(\frac{d}{dx}(x)=1\)
- \(\frac{d}{dx}(cx)=c\)
- \(\frac{d}{dx}(x^n)=nx^{n-1}\qquad n\in Q\)
- \(\frac{d}{dx}(u\pm v\pm w\pm...)=\frac{du}{dx}\pm \frac{dv}{dx}\pm \frac{dw}{dx}\pm ...\)
- Product rule $$ \frac{d}{dx}(f(x)\cdotp g(x))=f'(x)g(x)+f(x)g'(x) $$
- Quotient rule $$ \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2} $$
- Chain rule $$ \frac{dy}{dx}=\frac{dy}{du}\cdotp \frac{du}{dx} $$
- Reciprocal rule $$ \frac{d}{dx}\left(\frac{1}{f(x)}\right)=-\frac{f'(x)}{f(x)^2} $$
Derivate of Trigonometric Functions¶
- \(\frac{d}{dx}\sin x=\cos x\)
- \(\frac{d}{dx}\cos x=-\sin x\)
- \(\frac{d}{dx}\tan x=\sec^2 x\)
- \(\frac{d}{dx}\cot x=-\csc^2 x\)
- \(\frac{d}{dx}\csc x=-\csc x\cot x\)
- \(\frac{d}{dx}\sec x=\sec x\tan x\)
Derivate of Inverse Trigonometric Functions¶
- \(\frac{d}{dx}\sin^{-1}x=\frac{1}{\sqrt{1-x^2}}\qquad[x\in(-1,1)]\)
- \(\frac{d}{dx}\cos^{-1}x=-\frac{1}{\sqrt{1-x^2}}\qquad[x\in(-1,1)]\)
- \(\frac{d}{dx}\tan^{-1}x=\frac{1}{1+x^2}\qquad[x\in R]\)
- \(\frac{d}{dx}\cot^{-1}x=-\frac{1}{1+x^2}\qquad[x\in R]\)
- \(\frac{d}{dx}\csc^{-1}x=-\frac{1}{\sqrt{x^2-1}}\qquad\left[x\in[-1,1]'\right]\)
- \(\frac{d}{dx}\sec^{-1}x=\frac{1}{\sqrt{x^2-1}}\qquad\left[x\in[-1,1]'\right]\)
Derivative of Hyperbolic Functions¶
- \(\frac{d}{dx}\sinh x=\cosh x\)
- \(\frac{d}{dx}\cosh x=\sinh x\)
- \(\frac{d}{dx}\tanh x=\;\text{sech}^2 x\)
- \(\frac{d}{dx}\coth x=-\;\text{csch}^2 x\)
- \(\frac{d}{dx}\;\text{csch} x=-\;\text{csch} x\coth x\)
- \(\frac{d}{dx}\;\text{sech} x=-\;\text{sech} x\tanh x\)
Derivate of Inverse Hyperbolic Functions¶
- \(\frac{d}{dx}\sinh^{-1}x=\frac{1}{\sqrt{1+x^2}}\qquad x\in R\)
- \(\frac{d}{dx}\cosh^{-1}x=\frac{1}{\sqrt{x^2-1}}\qquad x>1\)
- \(\frac{d}{dx}\tanh^{-1}x=\frac{1}{1-x^2}\qquad|x|<1\)
- \(\frac{d}{dx}\coth^{-1}x=\frac{1}{1-x^2}\qquad|x|>1\)
- \(\frac{d}{dx}\;\text{csch}^{-1}x=-\frac{1}{\sqrt{1-x^2}}\qquad0<x<1\)
- \(\frac{d}{dx}\;\text{sech}^{-1}x=-\frac{1}{|x|\sqrt{1+x^2}}\qquad x\in R-{0}\)
Derivative of Exponential & Logarithmic Functions¶
- \(\frac{d}{dx}e^x=e^x\)
- \(\frac{d}{dx}e^{f(x)}=e^{f(x)}f'(x)\)
- \(\frac{d}{dx}a^x=a^x\ln a\)
- \(\frac{d}{dx}a^{f(x)}=a^{f(x)}\ln a f'(x)\)
- \(\frac{d}{dx}\ln x=\frac{1}{x}\)
- \(\frac{d}{dx}\log_a x=\frac{1}{x\ln a}\)
- \(\frac{d}{dx}\ln f(x)=\frac{1}{f(x)}f'(x)\)
Higher Derivative¶
- Second derivative: $$ \frac{d}{dx}\left(\frac{dy}{dx}\right)= \frac{d^2y }{dx^2} = f''(x) =y'' =D^2y $$
- Third derivative: $$ \frac{d}{dx}\left(\frac{d^2y }{dx^2}\right)= \frac{d^3y }{dx^3} = f'''(x) =y''' =D^3y $$
- nth derivative: $$ \frac{d}{dx}\left(\frac{d^{n-1}y }{dx^{n-1}}\right)= \frac{d^ny }{dxn} = f{(n)}(x) =y^{(n)} =D^ny $$
Series Expansions of Functions¶
Taylor series¶
If \(f\) is defined interval containing \(a\) and its derivatives of all orders exists at \(x=a\) then, Taylor series expansion of a function \(f(x)\) at \(x=a\) is: $$ f(x)=f(a)+(x-a)f'(a)+\frac{(x-a)^2f''(a)}{2!} + \frac{(x-a)^3f'''(a)}{3!}+... $$
\[ f(x)=\displaystyle\sum^{\infty}_{n=0}\frac{f^{(n)}(a)(x-a)^n}{n!} \]
Maclaurin Series¶
If \(f\) is defined in the interval containing \(0\) and its derivative exists at \(x=0\) then, Maclaurin series expansion of a function is:
\[ f(x)=f(0)+ xf'(0)+ \frac{f''(0)x^2}{2!} +\frac{f'''(0)x^3}{3!}+... \]
\[ f(x)=\sum^{\infty}_{n=0}\frac{f^{(n)}(0)x^n}{n!} \]
Important Maclaurin Series Expansions:¶
Function | Maclaurin Series |
---|---|
\(\sin x\) | \(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+....\) |
\(\cos x\) | \(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+....\) |
\(\tan x\) | \(x+\frac{x^3}{3}+\frac{2x^5}{15}+\frac{17x^7}{315}+....\) |
\(e^x\) | \(1+x-\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+....\) |
\(a^x\) | \(1+\ln a + \frac{x^2(\ln a)^2}{2!} +\frac{x^3 (\ln a)^3}{3!}+....\) |
Applications of derivatives¶
- Geometric Meaning of \(\frac{dy}{dx}\): If \(y=f(x)\) then \(\frac{dy}{dx}\) denotes the slope of the tangent to the given curve at any point \((x,y)\)
- Equation of Tangent to \(y=f(x)\) at \((x_1, y_1)\): $$ y-y_1=\left(\frac{dy}{dx}\right)_{(x_1,y_1)} (x-x_1) $$
- Eqation of Normal to \(y=f(x)\) at \((x_1,y_1)\): $$ y-y_1=\frac{-1}{\left(\frac{dy}{dx}\right)_{x_1,y_1}}(x-x_1) $$
- Increasing Function: Let a function \(f(x)\) is defined in the interval \((a,b)\), then \(f(x)\) is said to be increasing if
- \(f(x_1)>f(x_2)\) where \(x_1>x_2\)
- \(f'(x)>0\) where \(x\in(a,b)\)
- Extreme Values: Let \(f\) be a differentiable function in neighborhood of c where \(f'(c)=0\), then
- \(f\) has relative maxima at \(c\) if \(f''(c)<0\)
- \(f\) has relative minima at \(c\) if \(f''(c)>0\)
- Critical Point: The point \(c\) where \(f'(c)=0\) or undefined is called critical point.
- Point of Inflexion: The point \(c\) where \(f''(c)=0\) is called point of Inflexion.
- Differentials: Let \(y=f(x)\) is differentiable function then \(dy=f'(x)dx\) is called differential of the function.
- Stationary Point: Any point where f is neither increasing nor decreasing but \(f'(c)=0\) is called stationary point.