# How do you find the slope of the curve #f(x)=sqrt(x-1)# at the point x=5?

The slope is

(there are two possible values as the function is not one-one as it contains a square root)

The slope of a function at some particular value of the independent variable is the first derivative evaluated at that particular value.

Denoting the two component functions as

and

it might be noted that

The chain rule states

That is, the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

so that the rules of polynomial differentiation might be easily applied

evaluating this at the value of the inner function

Noting

The overall derivative is

That is

so

By signing up, you agree to our Terms of Service and Privacy Policy

To find the slope of the curve ( f(x) = \sqrt{x-1} ) at the point ( x = 5 ), you can use the derivative of the function. The derivative of ( f(x) ) with respect to ( x ) is ( f'(x) = \frac{1}{2\sqrt{x-1}} ). Evaluating this derivative at ( x = 5 ) gives:

[ f'(5) = \frac{1}{2\sqrt{5-1}} = \frac{1}{2\sqrt{4}} = \frac{1}{2 \times 2} = \frac{1}{4} ]

So, the slope of the curve ( f(x) = \sqrt{x-1} ) at the point ( x = 5 ) is ( \frac{1}{4} ).

By signing up, you agree to our Terms of Service and Privacy Policy

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.

- How do you use the limit definition to compute the derivative, f'(x), for #5x^2-3x+7#?
- How do you find the equation of the line tangent to the graph of #y = x^2 - 3# at the point P(2,1)?
- What is the equation of the tangent line of #f(x)=(5+4x)^2 # at #x=7#?
- How do you find the equation of a line tangent to #y=sqrtx# at (9,3)?
- What is the equation of the line tangent to #f(x)=(3x-4)/(x-2)^2# at #x=2#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7