# How do you differentiate #f(x) =((65e^-7x)+2)^3 # using the chain rule?

We have to find:

What I do after this is just expansion. It isn't needed, and it just makes it more confusing.

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To differentiate ( f(x) = ((65e^{-7x}) + 2)^3 ) using the chain rule, follow these steps:

- Let ( u = (65e^{-7x}) + 2 ).
- Find ( \frac{du}{dx} ).
- Differentiate ( u^3 ) with respect to ( u ).
- Multiply ( \frac{du}{dx} ) by the result from step 3.

Step-by-step calculation:

- Let ( u = (65e^{-7x}) + 2 ).
- ( \frac{du}{dx} = \frac{d}{dx}[(65e^{-7x}) + 2] ) ( = 65 \frac{d}{dx}(e^{-7x}) + 0 ) ( = 65 \cdot (-7e^{-7x}) ) ( = -455e^{-7x} ).
- Differentiate ( u^3 ) with respect to ( u ): ( \frac{d}{du}(u^3) = 3u^2 ).
- Multiply ( \frac{du}{dx} ) by the result from step 3: ( -455e^{-7x} \cdot (3((65e^{-7x}) + 2)^2) ).

So, the derivative of ( f(x) = ((65e^{-7x}) + 2)^3 ) with respect to ( x ) using the chain rule is ( -455e^{-7x} \cdot (3((65e^{-7x}) + 2)^2) ).

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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.

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