Home > Calculus > Derivatives of y=sec(x), y=cot(x), y= csc(x)

How do you find the derivative of #y = csc^3(2x) #?

Answer 1

Aurora Alvarado

#dy/dx=-6csc^3(2x)cot(2x)#

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

#"given "y=f(g(x))" then"#

#dy/dx=f'(g(x))xxg'(x)larrcolor(blue)"chain rule"#

#y=csc^3(2x)=(csc(2x))^3#

#dy/dx=3(csc(2x))^2xxd/dx(csc(2x))#

#color(white)(dy/dx)=3csc^2(2x)xx-csc(2x)cot(2x)xxd/dx(2x)#

#color(white)(dy/dx)=-6csc^3(2x)cot(2x)#

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Answer 2

Aurora Alvarado

To find the derivative of ( y = \csc^3(2x) ), you would use the chain rule. First, rewrite ( y ) as ( y = (\csc(2x))^3 ). Then, differentiate using the chain rule:

[ \frac{dy}{dx} = 3(\csc(2x))^2 \cdot (-\csc(2x) \cot(2x)) \cdot 2 ]

Simplify this expression to get the final derivative.

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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.