# How to Find The Slope of Roots: Derivative of Radicals

Derivatives of radicals, or root functions, often involve the power rule by expressing roots as fractional exponents. For instance, the square root of \( x\), is equal to \( x^{1/2} \) , making the power rule applicable. However, this method isn't always straightforward, especially with nested or complex radicals, where other techniques like the chain rule might be necessary. Understanding when to apply the power rule versus other methods is key in efficiently handling radical derivatives. But there are specific formulas that depend on the index of the radical and the exponent of the function under the radical.

## 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**

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

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