Class 11: Derivatives – Lecture Notes

$\displaystyle \textbf{1. Differentiability of a Function}$
$\displaystyle \text{A function }f(x)\text{ is differentiable at }x=c\text{ iff}$
$\displaystyle \lim_{x\to c}\frac{f(x)-f(c)}{x-c}$
$\displaystyle \text{exists finitely.}$
$\displaystyle \text{This limit is called the derivative or differentiation of }f(x)\text{ at }x=c\text{ and is denoted by }f'(c).$
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$\displaystyle \textbf{2. Geometrical Meaning of Derivative}$
$\displaystyle \text{Geometrically the derivative of a function }f(x)\text{ at a point }x=c\text{ is the slope of the tangent}$
$\displaystyle \text{to the curve }y=f(x)\text{ at the point }(c,f(c)).$
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$\displaystyle \textbf{3. Derivative with Respect to }x$
$\displaystyle \text{If }f(x)\text{ is a differentiable function, then}$
$\displaystyle \lim_{h\to0}\frac{f(x+h)-f(x)}{h}$
$\displaystyle \text{is called the differentiation of }f(x)\text{ with respect to }x.$
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$\displaystyle \textbf{4. Rate of Change}$
$\displaystyle \text{Mechanically, }\frac{d}{dx}[f(x)]\text{ measures the rate of change of }f(x)\text{ with respect to }x.$
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$\displaystyle \textbf{5. Standard Derivatives}$
$\displaystyle \text{(i) }\frac{d}{dx}(x^n)=nx^{n-1}$
$\displaystyle \text{(ii) }\frac{d}{dx}(a^x)=a^x\log_e a,\;a>0,\;a\neq1$
$\displaystyle \text{(iii) }\frac{d}{dx}(e^x)=e^x$
$\displaystyle \text{(iv) }\frac{d}{dx}(\log_e x)=\frac{1}{x}$
$\displaystyle \text{(v) }\frac{d}{dx}(\sin x)=\cos x$
$\displaystyle \text{(vi) }\frac{d}{dx}(\cos x)=-\sin x$
$\displaystyle \text{(vii) }\frac{d}{dx}(\tan x)=\sec^2 x$
$\displaystyle \text{(viii) }\frac{d}{dx}(\cot x)=-\mathrm{cosec}^2 x$
$\displaystyle \text{(ix) }\frac{d}{dx}(\sec x)=\sec x\tan x$
$\displaystyle \text{(x) }\frac{d}{dx}(\mathrm{cosec}\,x)=-\mathrm{cosec}\,x\cot x$
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$\displaystyle \textbf{6. Fundamental Rules of Differentiation}$
$\displaystyle \text{(i) Differentiation of a constant function is zero, i.e.,}$
$\displaystyle \frac{d}{dx}(c)=0$
$\displaystyle \text{(ii) Differentiation of a constant and a function is equal to constant times the differentiation}$
$\displaystyle \text{of the function.}$
$\displaystyle \text{(iii) If }f(x)\text{ and }g(x)\text{ are differentiable functions, then}$
$\displaystyle \text{(a) }\frac{d}{dx}\{f(x)\pm g(x)\}=\frac{d}{dx}(f(x))\pm\frac{d}{dx}(g(x))$
$\displaystyle \text{(b) }\frac{d}{dx}\{f(x)\times g(x)\}=g(x)\frac{d}{dx}(f(x))+f(x)\frac{d}{dx}(g(x))$
$\displaystyle \text{(c) }\frac{d}{dx}\left\{\frac{f(x)}{g(x)}\right\}=\frac{g(x)\frac{d}{dx}(f(x))-f(x)\frac{d}{dx}(g(x))}{\{g(x)\}^2}$
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