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

Take the derivative of the outside, and leave the inside alone, then multiply it by the derivative of the inside.

So...

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

- Identify the outer function and the inner function.
- Find the derivative of the outer function with respect to its inner function.
- Find the derivative of the inner function with respect to ( x ).
- Multiply the results of steps 2 and 3 together to get the final derivative.

Let's break it down:

- Outer function: ( g(u) = u^3 ), where ( u = 65e^{-7x} + 2 ).
- Derivative of the outer function: ( g'(u) = 3u^2 ).
- Inner function: ( u = 65e^{-7x} + 2 ).
- Derivative of the inner function: ( u' = \frac{d}{dx}(65e^{-7x} + 2) = -455e^{-7x} ).
- Multiply the derivatives: ( g'(u) \cdot u' = 3(65e^{-7x} + 2)^2 \cdot (-455e^{-7x}) ).
- Simplify the expression: ( \frac{d}{dx}((65e^{-7x} + 2)^3) = -1365(65e^{-7x} + 2)^2e^{-7x} ).

So, the derivative of ( f(x) = ((65e^{-7x}) + 2)^3 ) using the chain rule is ( -1365(65e^{-7x} + 2)^2e^{-7x} ).

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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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