There are rules or formulas that allow us to evaluate derivatives quickly without explicitly going through the limit process. We shall look at some rules in this section.

Rule. If c is a constant, then

d/dx (c) = 0

That is, the derivative of a constant function is zero.

Proof. Let f (x) = c. Aapplying the definition of the derivative, we obtain

f ' (x) = lim f (x+ h) - f (x)
         h→0

f ' (x) = lim c-c / h  = lim 0/h = 0
          h→0           h→0

For example, if y = 5, then dv/dx = d 5/dx = 0

Rule. If n is any real number, then d/dx (xⁿ) = n x^n-1

That is, the derivative of a constant power of x is the exponent times x raised to a power one less than the given power.

Proof. We shall give a proof for the case when is a positive integer. Let f (x) = xⁿ. By applying the definition of the derivative, we obtain

f ' (x) = lim  f (x+ h) - f (x)
         h→0        h

      = lim (x + h)ⁿ - xⁿ
       h→0    h

      = lim    (x+h)ⁿ - xⁿ
      x+h→x  (x + h) - x

       = n x^n-1         

Hence        f ‘ (c) = n x^n-1 .


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