How do you use the chain rule to differentiate #f(x)=((3x5)/(4x7))^6#?
This function can be differentiated using a combination of the "quotient rule" and the "chain rule".
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To differentiate ( f(x) = \left( \frac{3x  5}{4x  7} \right)^{6} ) using the chain rule, follow these steps:

Rewrite the function as ( f(x) = \left( g(x) \right)^{6} ), where ( g(x) = \frac{3x  5}{4x  7} ).

Find the derivative of the inner function ( g(x) ) with respect to ( x ), denoted as ( g'(x) ).

Apply the chain rule, which states that if ( y = u^n ), where ( u ) is a differentiable function of ( x ), then ( \frac{dy}{dx} = n \cdot u^{n1} \cdot \frac{du}{dx} ).

Substitute ( g(x) ) and ( g'(x) ) into the chain rule formula to find ( \frac{df}{dx} ), the derivative of ( f(x) ) with respect to ( x ).
So, ( \frac{df}{dx} = 6 \cdot \left( \frac{3x  5}{4x  7} \right)^{7} \cdot \frac{d}{dx} \left( \frac{3x  5}{4x  7} \right) ).
To find ( \frac{d}{dx} \left( \frac{3x  5}{4x  7} \right) ), use the quotient rule: ( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v  uv'}{v^2} ), where ( u = 3x  5 ) and ( v = 4x  7 ).
After calculating ( \frac{d}{dx} \left( \frac{3x  5}{4x  7} \right) ), substitute the result back into ( \frac{df}{dx} ) to get the final answer.
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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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