How to Find The Slope of Roots: Derivative of Radicals

Finding the derivative of simple radicals:

to find the derivative of  \( \sqrt{x} \) , we use:

\( \left(\sqrt{x}\right)’ = \frac{1}{2\sqrt{x}} \)

Which you already know.

Finding the derivative of radicals with functions inside them:

To find the derivative of \( \sqrt{f(x)} = (f(x))^{1/2} \):

\( \left(\sqrt{f(x)}\right)’ = \frac{1}{2}(f(x))^{-1/2} \cdot f'(x) \)

or

\( \left(\sqrt{f(x)}\right)’ = \frac{f'(x)}{2\sqrt{f(x)}} \)

Let’s solve an example: \( \text{Find the derivative of } \sqrt{x + 5}. \)

1. Simplify the function

\( f(x) = \sqrt{x + 5} = (x + 5)^{1/2} \)

2. Apply the power and chain rules

\( f'(x) = \frac{1}{2}(x + 5)^{-1/2} \cdot 1 = \frac{1}{2\sqrt{x + 5}} \)

Here is another example of this: \( \text{Find the derivative of } \sqrt{x^2 + 1}. \)

3. Simplify the function

\( f(x) = x^2 + 1 \)

4. Define the derivative of \(f(x)\)

\( f'(x) = 2x \)

5.Apply the formula

\( \left(\sqrt{x^2 + 1}\right)’ = \frac{2x}{2\sqrt{x^2 + 1}} = \frac{x}{\sqrt{x^2 + 1}} \)

Finding the derivative of radicals with functions inside them, with index of \( m\) :

To differentiate radicals with nested functions, use the chain rule and express the radical as a fractional exponent for simpler calculation.

As a general form, we have this formula:

\( \left(\sqrt[m]{f(x)}\right)’ = \frac{f'(x)}{m(f(x))^{1/m – 1}} \)

Consider this example: \( \text{Find the derivative of } \sqrt[3]{x^2}. \)

1. Define the function and its derivative

\( f(x) = x^2 \Rightarrow f'(x) = 2x \)

2. Apply the formula

\( \left(\sqrt[3]{x^2}\right)’ = \frac{2x}{3(x^2)^{2/3}} = \frac{2x}{3x^{4/3}} \)

Here is a more complex problem: \( \text{Find the derivative of } \sqrt[4]{x^3 + 2x}. \)

3. Define the function and its derivative

\( f(x) = x^3 + 2x \Rightarrow f'(x) = 3x^2 + 2 \)

4. Apply the formula

\( \left(\sqrt[4]{x^3 + 2x}\right)’ = \frac{3x^2 + 2}{4(x^3 + 2x)^{3/4}} \)