# How do you find the derivative of the inverse of #f(x)=7x+6# using the definition of the derivative of an inverse function?

Please see the explanation section below.

I'm not sure what you mean by the "definition of the derivative of an inverse function".

The definition of the derivative of any function is the same. We can write the definition using inverse function notation as

but that's just notation.

I'm guessing that you want to use a Theorem about derivatives of inverse functions.

In this case

Therefore

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To find the derivative of the inverse of ( f(x) = 7x + 6 ) using the definition of the derivative of an inverse function, follow these steps:

- Let ( y = f(x) = 7x + 6 ).
- Swap x and y to express x in terms of y: ( x = \frac{{y - 6}}{7} ).
- Differentiate both sides of the equation with respect to y to find ( \frac{{dx}}{{dy}} ).
- Use the chain rule to express ( \frac{{dx}}{{dy}} ) in terms of ( \frac{{dy}}{{dx}} ).
- Solve for ( \frac{{dy}}{{dx}} ) to find the derivative of the inverse function.

The derivative of the inverse of ( f(x) = 7x + 6 ) using the definition of the derivative of an inverse function is:

[ \frac{{d}}{{dy}} \left( \frac{{y - 6}}{7} \right) = \frac{1}{7} ]

Therefore, the derivative of the inverse function is ( \frac{1}{7} ).

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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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