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What is the derivative of #sin^5(x)#?

Answer 1

Evelyn Agard

#5sin^4xcosx#

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

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

let #y=sin^5x=(sinx)^5#

let #u=sinxrArr(du)/(dx)=cosx#

and #y=u^5rArr(dy)/(du)=5u^4#

substitute these values into (A) convert u back to x

#rArrdy/dx=5u^4(cosx)=5sin^4xcosx#

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Answer 2

Luke Alvarez

The derivative of ( \sin^5(x) ) with respect to ( x ) is ( 5\sin^4(x) \cos(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.