How do you find the derivative of #f(x)=(5x^6sqrt x) + (3/(x^3 sqrt x))#?

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Answer 1 · 2016-07-22T01:33:50

#f'(x) = 1/2(65x^5sqrt(x) -21/(x^4sqrt(x)))#

#f(x) = (5x^6sqrt(x)) + (3/(x^3sqrt(x)))#

Using the rules of indicies #f(x)# can be written:

#f(x) =5x^6x^(1/2) + 3x^-3x^(-1/2)# #= 5x^(13/2) + 3x^(-7/2)#

Aplying the Power Rule to both terms:

#f'(x) = 5* 13/2 x^(13/2-1) + 3* (-7/2) x^(-7/2-1)#
#= 1/2(65x^(11/2) -21x^(-9/2))#

To express #f'(x)# in the form of #f(x)# in the original question, we can rewrite #f'(x)# as:
#f'(x) = 1/2(65x^5 * x^(1/2) - 21x^(-4) * x^(-1/2))#

#=1/2(65x^5 sqrt(x) - 21/(x^4 sqrt(x)))#