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More examples of derivatives

Here are some more examples of derivatives of functions, obtained using the first principles of differentiation.

 

Example 1: Find the derivative of f(x) = 6 using first principles.

f(x + h)  =  6

f'(x)  =  lim{h right 0}~{{f(x+h)~-~f(x)}/{h}}

=  lim{h right 0}~{{6~-~6}/{h}}

= 0 (as h <>0)

Remember: In general if f(x) = c, where c is a fixed real number, then f'(x) = 0.

 

Example 2: Find the derivative of f(x) = x3

f(x + h)  =  (x + h)3

=  x3 + 3x2h + 3xh2 + h3

f(x + h) – f(x)  =  (x3 + 3x2h + 3xh2 + h3) – x3

=  3x2h + 3xh2 + h3

f'(x)  =  lim{h right 0}~{{f(x+h)~-~f(x)}/{h}}

=   lim{h right 0}~{{3x^2h~+~3xh^2~+~h^3}/{h}}

=   lim{h right 0}~{{3x^2~+~3xh~+~h^2}}

=  3x2

Remember: If we have an equation with power in it, the derivative of the equation reduces the power index by 1, and the function’s power becomes the co-efficient of the derivative function.

In other words, if f(x)  =  xn, then f'(x)  =  nxn-1

The above generalisation will hold for negative powers also.

 

Example 3: Find the derivative of f(x) =  1/x

f(x + h)  =    1/{x+h}

f'(x)  =  lim{h right 0}~{{f(x+h)~-~f(x)}/{h}}

=   lim{h right 0}~{{1/{x+h}~-~1/x}/{h}}

=   lim{h right 0}~{{x~-~(x+h)}/{x(x+h)h}}

=   lim{h right 0}~{{-h}/{x(x+h)h}}

=   lim{h right 0}~{{-1}/{x(x+h)}}

=  -~1/x^2

So derivative of f(x)  =  -~1/x^2

 

Example 4: Find the derivative of f(x) =  1/x^2

f(x + h)  =    1/{(x+h)^2}

f'(x)  =  lim{h right 0}~{{f(x+h)~-~f(x)}/{h}}

=   lim{h right 0}~{{1/{(x+h)^2}~-~1/x^2}/{h}}

=   lim{h right 0}~{{x^2~-~(x+h)^2}/{x^2(x+h)^2h}}

=   lim{h right 0}~{{(x-x-h)(x+x+h)}/{x^2(x+h)^2h}}  using a2 – b2 = (a-b)(a+b)

=   lim{h right 0}~{{-h(2x+h)}/{x^2(x+h)^2h}}

=   – lim{h right 0}~{{(2x+h)}/{x^2(x+h)^2}}

=   – {{2x}/{x^2(x^2)}}

=   – {{2}/{x^3}}

So when f(x) =  1/x^2 ,   f'(x) =  -2/x^3

Remember: In general, if f(x) = 1/x^n, f'(x) = -n/x^{n+1}

and, if f(x) = x-n, f'(x)  =  -n(x)-n-1  =  -n/x^{n+1}

 

To summarise

  1. If f(x) = c, where c is a fixed real number, then f'(x) = 0.
  2. If f(x)  =  xn, then f'(x)  =  nxn-1
  3. If f(x) = 1/x^n, f'(x) = -n/x^{n+1}
  4. If f(x) = x-n, f'(x)  =  -n(x)-n-1  =  {-n}/x^{n+1}

 

We now look at finding derivatives using formula