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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