Today we just started learning about derivatives with trig, and I need help with two questions. I'm confused on how to work the denominator, and then for number two I have no idea what to do....

Today we just started learning about derivatives with trig, and I need help with two questions. I'm confused on how to work the denominator, and then for number two I have no idea what to do. Please Help!

1) Differentiate: y=x/(sinx+cosx)

=1(sosx-(-sinx))-x(sinx+cosx)/(sinx+cosx)^2

=cosx+sinx-xsinx-xcosx/(sinx+cosx)^2

2) Differentiate: H(x)=csc(x)+e^x(cot(x))

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embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

(1) Differentiate `y=x/(sinx+cosx)` :

Use the quotient rule: `y=f/(g) ==> y'=(f'g-fg')/g^2`

`y'=(1(sinx+cosx)-x(cosx-sinx))/(sinx+cosx)^2`

`=(sinx+xsinx+cosx-xcosx)/((sinx+cosx)^2)`

`=(sinx(1+x)+cosx(1-x))/(sinx+cosx)^2`

(2) Differentiate `H(x)=cscx+e^x cotx`

Use the product rule:

`H'(x)=-cscxcotx+(e^xcotx+e^x(-csc^2x))`

`=-cscxcotx+e^x(cotx-csc^2x)`

`=-cscxcotx+e^x(cotx-(1+cot^2x))`

`=-cscxcotx+e^x(-cot^2x+cotx-1)`

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