Question

Use the definition of the derivative to compute a formula for the derivative of the given function. Use the formula to determine the slope of the line tangent to the graph of the function at the indicated value. f(x) =√x at x = 25

Answer 1

So, the derivative of the function
`\(f(x) = √(x)\) at \(x = 25\) is \(f'(25) = (1)/(10)\).`

Now, we have the slope of the tangent line to the graph of the function at
`\(x = 25\)`. The slope is
`\((1)/(10)\)`.

Explanation:

To find the derivative of the function
`\(f(x) = √(x)\) at \(x = 25\)`, we'll use the definition of the derivative:

`\[f'(x) = \lim_(h \to 0) (f(x + h) - f(x))/(h)\]`

`First, plug in the function \(f(x) = √(x)\):`

`\[f'(x) = \lim_(h \to 0) (√(x + h) - √(x))/(h)\]`

Now, we'll calculate the derivative using this limit:

`\[f'(x) = \lim_(h \to 0) (√(x + h) - √(x))/(h) \cdot (√(x + h) + √(x))/(√(x + h) + √(x))\]`

This step involves multiplying the numerator and denominator by the conjugate of the numerator to eliminate the square root in the numerator. This is a common technique when dealing with square root functions.

Simplify the expression:

`\[f'(x) = \lim_(h \to 0) ((x + h) - x)/(h(√(x + h) + √(x)))\]`

Now, simplify the numerator:

`\[f'(x) = \lim_(h \to 0) (h)/(h(√(x + h) + √(x)))\]`

Notice that we can cancel \(h\) in the numerator and denominator:

`\[f'(x) = \lim_(h \to 0) (1)/(√(x + h) + √(x))\]`

Now, evaluate the limit as
`\(h\)` approaches 0:

`\[f'(25) = (1)/(√(25 + 0) + √(25))\]`

`\[f'(25) = (1)/(√(25) + √(25))\]`

`\[f'(25) = (1)/(5 + 5)\]`

`\[f'(25) = (1)/(10)\]`

So, the derivative of the function
`\(f(x) = √(x)\) at \(x = 25\) is \(f'(25) = (1)/(10)\).`

Now, we have the slope of the tangent line to the graph of the function at
`\(x = 25\)`. The slope is
`\((1)/(10)\)`.