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Question
Find the derivative of f(x) = `sqrt(sinx)`, by first principle.
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Solution
By definition,
f'(x) = `lim_(h -> 0) (f(x + h) - f(x))/h`
= `lim_(h -> 0) (sqrt(sin (x + h)) - sqrt(sin x))/h`
= `lim_(h -> 0) ((sqrt(sin(x + h)) - sqrt(sinx))(sqrt(sin(x + h)) + sqrt(sinx)))/(h(sqrt(sin(x + h)) + sqrt(sinx))`
= `lim_(h -> 0) (sin(x + h) - sinx)/(h(sqrt(sin(x + h)) + sqrt(sinx))`
= `lim_(h -> 0) (2 cos (2x + h)/2 sin h/2)/(2 * h/2 (sqrt(sin(x + h)) + sqrt(sinx))`
= `cosx/(2sqrt(sinx))`
= `1/2 cot x sqrt(sinx)`
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