How do you use the chain rule to differentiate #y=(7-x)^4#?
Simplify:
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To differentiate ( y = (7 - x)^4 ) using the chain rule, follow these steps:
- Identify the outer function, which is ( u = x^4 ).
- Identify the inner function, which is ( v = 7 - x ).
- Differentiate the outer function with respect to its variable, which is ( u ) with respect to ( x ), resulting in ( \frac{du}{dx} = 4x^3 ).
- Differentiate the inner function with respect to its variable, which is ( v ) with respect to ( x ), resulting in ( \frac{dv}{dx} = -1 ).
- Apply the chain rule formula: ( \frac{dy}{dx} = \frac{du}{dv} \times \frac{dv}{dx} ).
- Substitute ( \frac{du}{dx} ) and ( \frac{dv}{dx} ) into the chain rule formula.
- Simplify to get the derivative of ( y ) with respect to ( x ).
Applying these steps:
( \frac{dy}{dx} = \frac{du}{dv} \times \frac{dv}{dx} ) ( = \frac{du}{dx} \times \frac{dv}{dx} ) ( = (4x^3) \times (-1) ) ( = -4x^3 )
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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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