# Calculus Solutions, Important Examples, Formulas and Videos

## Theorem

• Derivative of sum of two functions is sum of the derivatives of the function.
$\frac{\mathrm{d} }{\mathrm{d}x}\left [ f(x)+g(x) \right ]=\frac{\mathrm{d} }{\mathrm{d} x}f(x)+\frac{\mathrm{d} }{\mathrm{d} x}g(x)$
• Derivative of difference of two functions is difference of the derivatives of the function.
$\frac{\mathrm{d} }{\mathrm{d}x}\left [ f(x)-g(x) \right ]=\frac{\mathrm{d} }{\mathrm{d} x}f(x)-\frac{\mathrm{d} }{\mathrm{d} x}g(x)$
• Derivative of product of two functions is given by the following product rule.
$\frac{\mathrm{d} }{\mathrm{d}x}\left [ f(x).g(x) \right ]=\frac{\mathrm{d} }{\mathrm{d} x}f(x).\frac{\mathrm{d} }{\mathrm{d} x}g(x)$ .
• Derivatives of quotient of two function is given by the following rules (whenever the dimension is non -zero )
$\frac{\mathrm{d} }{\mathrm{d} x}\left ( \frac{f(x)}{g(x)} \right )=\frac{\frac{\mathrm{d} f(x)\cdot g(x)}{\mathrm{d} x}-f(x)\frac{\mathrm{d} }{\mathrm{d} x}g(x)}{g(x)^{2}}$

Derivatives of $f(x)=x^{n}\: is\: nx^{n-1}$ for any positive integer n . By definition we have
$f^{'}x=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=\lim_{h\rightarrow 0}\frac{(x+h)^{n}-x^{n}}{h}$

Derivatives of polynomial trigonometric functions : We start with the following theorem which tells us the derivative of a polynomial function .

Example: Let $h(x)=\frac{x^{5}-\cos x}{\sin x}$ We use the quotient rule on this function whereve.
$h^{'}x=\frac{(x^{5}-\cos x)^{'}\sin x-(x^{5}-\cos x)(\sin x)^{'}}{(\sin x)^{2}}=\frac{(5x^{4}+\sin x)(\sin x)-(x^{5}-\cos x)\cos x}{\sin ^{2}x}=\frac{-x^{5}\cos x+5x^{4}\sin x+1}{(\sin x^{2})}$

Points to be Remember:

• The expected value of the function as dictated by the points to the left of a point defines the left hand limit of the function at that point. Similarly the right hand limit .
• Limit of a function at a point is the common value of the left and right hand limit if they coincide.
• For a function f and a real number a , $\lim_{x\rightarrow a }f(x)$ and f(a) may not be same (in fact , one may be define and not other one )
• For function f and g the following hold:
$\lim_{x\rightarrow a}[f(x)+g(x)]=\lim_{x\rightarrow a}f(x)+\lim_{x\rightarrow 0}g(x)$
$lim_{x\rightarrow a}[f(x)-g(x)]=\lim_{x\rightarrow a}f(x)-\lim_{x\rightarrow 0}g(x)$
$\lim_{x\rightarrow a}[f(x)\cdot g(x)]=\lim_{x\rightarrow a}f(x)\cdot\lim_{x\rightarrow 0}g(x)$
$\lim_{x\rightarrow a}\frac{f(x)}{g(x)}=\frac{\lim_{x\rightarrow 0}f(x)}{\lim_{x\rightarrow a}g(x)}$
• Following are some of the standard limit
$\lim_{x\rightarrow a}\frac{x^{n}-a^{n}}{x-1}=na^{n-1}$
$\lim_{x\rightarrow a}\frac{\sin x}{x}=1$
$\lim_{x\rightarrow a}\frac{1-\cos x}{x}=0$
• The derivative of a function f at a is defined by
by $\lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h}$
• Derivative of a function f at any point x is defined by
$f^{'}x=\frac{\mathrm{d} f(x)}{\mathrm{d} x}=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$
• For functions u and v the following holds
$(u\pm v)^{'}=u^{'}\pm v^{'}$
$(u v)^{'}=u^{'}v+ v^{'}u$
$(\frac{u}{v})^{'}=\frac{u^{'}v-uv^{'}}{v^{2}}$ provided all are defined .
• Following are the some of the standard derivatives .
$\frac{\mathrm{d} }{\mathrm{d} x}(x)^{n}=nx^{n-1}$
$\frac{\mathrm{d} }{\mathrm{d} x}(\sin x)=\cos x$
$\frac{\mathrm{d} }{\mathrm{d} x}(\cos x)=-\sin x$

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