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How do you find the derivative of #y=x^5/(4sinx)#?

Answer 1

Paisley Johnson

#dy/dx=(x^4(5sinx-xcosx))/(4sin^2x)#

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

#"Given "y=(g(x))/(h(x))" then"#

#dy/dx=(h(x)g'(x)-g(x)h'(x))/(h(x))^2larrcolor(blue)"quotient rule"#

#g(x)=x^5rArrg'(x)=5x^4#

#h(x)=4sinxrArrh'(x)=4cosx#

#rArrdy/dx=(20x^4sinx-4x^5cosx)/(4sinx)^2#

#color(white)(rArrdy/dx)=(4x^4(5sinx-xcosx))/(16sin^2x)#

#color(white)(rArrdy/dx)=(x^4(5sinx-xcosx))/(4sin^2x)#

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Answer 2

Ezekiel Alexander

To find the derivative of ( y = \frac{x^5}{4\sin(x)} ), you would use the quotient rule, which states that if ( y = \frac{u}{v} ), then ( y' = \frac{u'v - uv'}{v^2} ), where ( u ) and ( v ) are functions of ( x ). In this case, ( u = x^5 ) and ( v = 4\sin(x) ). Differentiate ( u ) and ( v ) separately and then apply the quotient rule to find ( y' ).

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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.