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How do you differentiate #f(x)=(1+cos^2x)^6#?

Answer 1

Christopher Agresta

#f'(x)=-6sin2x(1+cos^2x)^5#

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

#"Given " f(x)=g(h(x))" then"#

#color(red)(bar(ul(|color(white)(2/2)color(black)(f'(x)=g'(h(x))xxh'(x))color(white)(2/2)|)))#

#rArrf'(x)=6(1+cos^2x)^5xxd/dx(1+cos^2x)to(1)#

#"Using the " color(blue)"chain rule " "on " (1+cos^2x)#

#d/dx(1+cos^2x)=2cosxxd/dx(cosx)#

#color(white)(d/dx(1+cos^2x))=-2cosxsinxlarr( -sin2x)#

#"Returning to " (1)#

#f'(x)=-6sin2x(1+cos^2x)^5#

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Answer 2

Isabella Ackerman

To differentiate the function ( f(x) = (1 + \cos^2(x))^6 ), you can use the chain rule along with the power rule. First, rewrite the function as ( f(x) = (1 + (\cos(x))^2)^6 ). Then, differentiate it term by term:

The outer function is ( u^6 ), where ( u = 1 + (\cos(x))^2 ).
The derivative of ( u^6 ) with respect to ( u ) is ( 6u^5 ).
The derivative of ( 1 + (\cos(x))^2 ) with respect to ( x ) is ( -2\cos(x)\sin(x) ).

Combine these results using the chain rule:

[ \frac{d}{dx} \left[ (1 + \cos^2(x))^6 \right] = 6(1 + \cos^2(x))^5 \cdot (-2\cos(x)\sin(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.