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"As" \ x \ "is the variable of" \ f(x), "I take it that the variable meant in" \ cos \ "is" \ x, "not" \ \theta. "If this is the case, we proceed as follows:"
"We are given:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad f(x) \ = \ 6 \sqrt{x} + 5 cos( x ).
"First rewrite" \ f(x):
\qquad \qquad \qquad \qquad \qquad \qquad \qquad f(x) \ = \ 6 x^{1/2} + 5 cos( x ).
"Now use the Sum Rule, and then the rules for the basic functions"
"present there:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad f'(x) \ = \ [ 6 x^{1/2} ]' + [ 5 cos( x ) ]'
\qquad \qquad \qquad \qquad \qquad \qquad \qquad f'(x) \ = \ 6 [ x^{1/2} ]' + 5 [ cos( x ) ]'
\qquad \qquad \qquad \qquad \qquad \qquad \qquad f'(x) \ = \ 6 [ 1/2 x^{- 1/2} ] + 5 ( -sin( x ) )
\qquad \qquad \qquad \qquad \qquad \qquad \qquad f'(x) \ = \ 6 \cdot 1/2 x^{- 1/2} - 5 sin( x )
\qquad \qquad \qquad \qquad \qquad \qquad \qquad f'(x) \ = \ 3 x^{- 1/2} - 5 sin( x ).
"Now remove the negative exponents, and write the fractional"
"exponent as a radical here:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad f'(x) \ = \ 3 ( 1/ x^{ 1/2} ) - 5 sin( x )
\qquad \qquad \quad :. \qquad \qquad \qquad f'(x) \ = \ 3 / \sqrt{x} - 5 sin( x ).
"This is our answer."
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"Summarizing:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad f(x) \ = \ 6 \sqrt{x} + 5 cos( x ).
\qquad \qquad \qquad \qquad \qquad \qquad \quad f'(x) \ = \ 3 / \sqrt{x} - 5 sin( x ).