How do you use the chain rule to differentiate #y=4(x^2+1)^2#?

Answer 1

#dy/dx =16x(x^2+1)#

#y=4(x^2+1)^2#
The chain rule states that: #d/dx (f(g(x)) = f'(g(x))*g'(x)#
Here: #f(x) = 4(g(x))^2# where #g(x)=x^2+1#

Applying the chain rule:

#dy/dx=4*2(x^2+1) * d/dx(x^2+1)#
#=8(x^2+1)*2x#
#=16x(x^2+1)#
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Answer 2

To differentiate ( y = 4(x^2+1)^2 ) using the chain rule, first, identify the outer function and the inner function. In this case, the outer function is ( u^2 ) and the inner function is ( x^2+1 ). Then, differentiate the outer function with respect to the inner function, and multiply it by the derivative of the inner function with respect to ( x ). The derivative of ( u^2 ) with respect to ( u ) is ( 2u ), and the derivative of ( x^2+1 ) with respect to ( x ) is ( 2x ). Thus, applying the chain rule, the derivative of ( y ) with respect to ( x ) is ( 8(x^2+1)(2x) ). Simplifying, we get ( 16x(x^2+1) ).

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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.

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.

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.

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.

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