Question

Question #4ab09

Answer 1

`1/4`

Explanation:

The slope of a function's tangent line at a point is given by the function's derivative. So, we need to find the derivative of `sqrt(x-5)=(x-5)^(1/2)`.

Note that `d/dxx^(1/2)=1/2x^(-1/2)`. So, the chain rule states that if we have a function to the `1//2` power instead, we follow the same process of the power rule, by moving down the `1//2` as a multiplicative constant and reducing the power by `1` `(`here, to get to `-1//2)`, but we also multiply by the derivative of the inner function.

Thus, `d/dx(f(x))^(1/2)=1/2(f(x))^(-1/2)*f'(x)`.

So, the derivative of `(x-5)^(1/2)` is:

`d/dx(x-5)^(1/2)=1/2(x-5)^(-1/2)*d/dx(x-5)`

The derivative of `x-5` is just `1`, so:

`dy/dx=1/2(x-5)^(-1/2)=1/(2sqrt(x-5))`

The slope of the tangent line to the original function `y` is found by plugging `x=9` into `dy//dx`:

`1/(2sqrt(9-5))=color(red)(1/4`