How do you use the chain rule to differentiate #y=(3x+7)^5#?
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To differentiate ( y = (3x + 7)^5 ) using the chain rule:

Identify the outer function ( u ) and the inner function ( v ). Let ( u = x^5 ) and ( v = 3x + 7 ).

Calculate the derivative of the outer function with respect to the inner function, ( \frac{du}{dv} ). ( \frac{du}{dv} = 5u^{4} ).

Calculate the derivative of the inner function with respect to ( x ), ( \frac{dv}{dx} ). ( \frac{dv}{dx} = 3 ).

Apply the chain rule formula: ( \frac{dy}{dx} = \frac{du}{dv} \cdot \frac{dv}{dx} ). ( \frac{dy}{dx} = 5(3x + 7)^4 \cdot (3) ).

Simplify the expression. ( \frac{dy}{dx} = 15(3x + 7)^4 ).
Therefore, the derivative of ( y = (3x + 7)^5 ) with respect to ( x ) is ( 15(3x + 7)^4 ).
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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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