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How do you differentiate #f(x)=(x^2+1)^3 # using the chain rule?

Answer 1

Aurora Alvarado

#f'(x)=6x(x^2+1)^2#

#f'(x)=3(x^2+1)^2+2*x#

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Answer 2

William Abdalla

#f'(x)=6x(x^2+1)^2#

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

#f'(x)=g'(h(x))xxh'(x)larrcolor(blue)"chain rule"#

#f'(x)=3(x^2+1)^2xxd/dx(x^2+1)#

#color(white)(f'(x))=3(x^2+1)^2xx2x=6x(x^2+1)^2#

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Answer 3

Paisley Johnson

To differentiate ( f(x) = (x^2 + 1)^3 ) using the chain rule:

Identify the outer function ( u ) and the inner function ( v ). Let ( u = (v)^3 ) and ( v = x^2 + 1 ).
Differentiate the outer function with respect to the inner function. ( \frac{du}{dv} = 3v^2 ).
Differentiate the inner function with respect to ( x ). ( \frac{dv}{dx} = 2x ).
Apply the chain rule: ( \frac{df}{dx} = \frac{du}{dv} \cdot \frac{dv}{dx} = 3(v^2) \cdot (2x) ).
Substitute the expressions for ( u ) and ( v ): ( \frac{df}{dx} = 3(x^2 + 1)^2 \cdot (2x) ).

Therefore, ( \frac{df}{dx} = 6x(x^2 + 1)^2 ).

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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.