How do you use the chain rule to differentiate #y=(x+1)^6/(3x2)^5#?
You avoid the quotient rule, and take the derivative using the product rule. The chain rule states:
So, you incorporate the nested function, and take the derivative of that function as well.
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To differentiate ( y = \frac{(x+1)^6}{(3x2)^5} ) using the chain rule, follow these steps:

Identify the inner and outer functions.
 Inner function: ( u = x + 1 )
 Outer function: ( y = u^6 )

Compute the derivatives of the inner and outer functions.
 ( \frac{du}{dx} = 1 )
 ( \frac{dy}{du} = 6u^5 )

Apply the chain rule:
 ( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} )

Substitute the derivatives and the inner function into the chain rule formula:
 ( \frac{dy}{dx} = 6u^5 \times 1 )

Replace ( u ) with the inner function ( x + 1 ):
 ( \frac{dy}{dx} = 6(x + 1)^5 )

Repeat the process for the denominator.
 Inner function: ( v = 3x  2 )
 Outer function: ( y = v^{5} )

Compute the derivatives of the inner and outer functions.
 ( \frac{dv}{dx} = 3 )
 ( \frac{dy}{dv} = 5v^{6} )

Apply the chain rule:
 ( \frac{dy}{dx} = \frac{dy}{dv} \times \frac{dv}{dx} )

Substitute the derivatives and the inner function into the chain rule formula:
 ( \frac{dy}{dx} = 5v^{6} \times 3 )

Replace ( v ) with the inner function ( 3x  2 ):
 ( \frac{dy}{dx} = 15(3x  2)^{6} )
Therefore, the derivative of ( y = \frac{(x+1)^6}{(3x2)^5} ) with respect to ( x ) is: [ \frac{dy}{dx} = 6(x + 1)^5 \cdot 15(3x  2)^{6} ]
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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