Home > Calculus > Intuitive Approach to the derivative of y=sin(x)

What is the derivative of this function #sin x^(1/2)#?

Answer 1

#cos(x^(1/2))/(2x^(1/2)#

differentiate using the #color(blue)"chain rule"#

#color(orange)"Reminder " color(red)(bar(ul(|color(white)(a/a)color(black)(dy/dx=(dy)/(du)xx(du)/(dx))color(white)(a/a)|)))....(A)#

let #u=x^(1/2)rArr(du)/(dx)=1/2x^(-1/2)#

and #y=sin(u)rArr(dy)/(du)=cos(u)#

replace these values with (A), converting u to terms of x

#rArrdy/dx=cos(u)xx1/2x^(-1/2)=cos(x^(1/2))/(2x^(1/2)#

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Answer 2

Paisley Johnson

The derivative of the function ( \sin(x^{1/2}) ) is ( \frac{1}{2\sqrt{x}} \cdot \cos(\sqrt{x}) ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.