# Nakafa Learning Content

> For AI agents: use [llms.txt](https://nakafa.com/llms.txt) for the site index. Markdown versions are available by appending `.md` to content URLs or sending `Accept: text/markdown`.

URL: https://nakafa.com/en/subjects/mathematics/derivative-function/derivative-of-algebraic-function
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/derivative-function/derivative-of-algebraic-function/en.mdx

Learn differentiation of algebraic functions using power, product, and quotient rules. Learn to differentiate polynomials, radicals, and rational functions.

---

## Applying the Derivative Rules

Now that we have mastered the various [properties of derivatives](/en/subjects/mathematics/derivative-function/properties-of-derivative-function), it's time to apply them to different forms of algebraic functions. Whether it's a polynomial, a rational function, or one containing a radical, the key is to recognize the function's structure and choose the right 'tool' or property to differentiate it. Let's see how this strategy is applied in a few examples.

## Using the Properties of Derivatives

Let's see how the properties of derivatives work in practice through a few examples.

### Differentiating a Polynomial

Find the first derivative of $$y = 4x^2 + x + 2$$.

Visible text: Find the first derivative of .

**Solution:**

We can differentiate each term one by one using the power rule and the constant rule.

Component: MathContainer
Children:

```math
y' = (4 \cdot 2)x^{2-1} + (1 \cdot 1)x^{1-1} + 0
```

```math
y' = 8x^1 + 1x^0
```

```math
y' = 8x + 1 \quad (\text{since } x^0=1)
```

### Conquering Radical Forms

Find the first derivative of $$y = x\sqrt{x^3}$$.

Visible text: Find the first derivative of .

**Solution:**

There are two ways to solve this.

**Method $$1$$: Using the Product Rule**

Visible text: **Method : Using the Product Rule**

First, we convert the radical form to an exponent: $$y = x \cdot x^{3/2}$$.

Visible text: First, we convert the radical form to an exponent: .

Let $$u(x) = x$$ and $$v(x) = x^{3/2}$$. Then, <InlineMath math="u'(x)=1" /> and <InlineMath math="v'(x) = \frac{3}{2}x^{1/2}" />.

Visible text: Let and . Then, <InlineMath math="u'(x)=1" /> and <InlineMath math="v'(x) = \frac{3}{2}x^{1/2}" />.

Component: MathContainer
Children:

```math
y' = u'(x)v(x) + u(x)v'(x)
```

```math
y' = (1)(x^{3/2}) + (x)(\frac{3}{2}x^{1/2})
```

```math
y' = x^{3/2} + \frac{3}{2}x^{3/2}
```

```math
y' = \frac{5}{2}x^{3/2}
```

**Method $$2$$: Simplifying First**

Visible text: **Method : Simplifying First**

We can simplify the function before differentiating it.

Component: MathContainer
Children:

```math
y = x \cdot x^{3/2} = x^{1 + 3/2} = x^{5/2}
```

```math
y' = \frac{5}{2}x^{5/2 - 1} = \frac{5}{2}x^{3/2}
```

> Both methods give the same result. Sometimes, simplifying the function first can make the differentiation process quicker.

### Tackling Rational Functions

Find the first derivative of $$y = 2\left(\frac{x^2+2}{x}\right)$$.

Visible text: Find the first derivative of .

**Solution:**

We use the quotient rule. Let $$u(x) = x^2+2$$ and $$v(x) = x$$. Then <InlineMath math="u'(x) = 2x" /> and <InlineMath math="v'(x) = 1" />.

Visible text: We use the quotient rule. Let and . Then <InlineMath math="u'(x) = 2x" /> and <InlineMath math="v'(x) = 1" />.

Component: MathContainer
Children:

```math
y' = 2 \cdot \left(\frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}\right)
```

```math
y' = 2 \cdot \left(\frac{(2x)(x) - (x^2+2)(1)}{x^2}\right)
```

```math
y' = 2 \cdot \left(\frac{2x^2 - x^2 - 2}{x^2}\right)
```

```math
y' = 2 \cdot \left(\frac{x^2 - 2}{x^2}\right)
```

```math
y' = 2 \cdot \left(\frac{x^2}{x^2} - \frac{2}{x^2}\right) = 2 \cdot \left(1 - \frac{2}{x^2}\right) = 2 - \frac{4}{x^2}
```

## Exercises

1.  Find the first derivative of $$y = \frac{\sqrt{x} + x^2}{2x}$$.

Visible text: 1. Find the first derivative of .

### Answer Key

1.  For this problem, the easiest way is to simplify the function before differentiating it.

    **Step** $$1$$: Split the Fraction

    We can break the fraction into two separate parts to make it easier.

    
    
    ```math
    y = \frac{\sqrt{x}}{2x} + \frac{x^2}{2x}
    ```

    **Step** $$2$$: Simplify Each Term

    Convert the square root to the exponent $$x^{1/2}$$ and use exponent properties to simplify each term.

    <MathContainer>
      
    
    ```math
    y = \frac{x^{1/2}}{2x^1} + \frac{x^2}{2x^1}
    ```

      
    
    ```math
    y = \frac{1}{2}x^{1/2 - 1} + \frac{1}{2}x^{2-1}
    ```

      
    
    ```math
    y = \frac{1}{2}x^{-1/2} + \frac{1}{2}x
    ```

    </MathContainer>

    **Step** $$3$$: Apply the Power Rule

    Once the function is simplified, we can directly differentiate it term by term.

    <MathContainer>
      <BlockMath math="y' = \frac{1}{2}\left(-\frac{1}{2}x^{-1/2 - 1}\right) + \frac{1}{2}(1)" />
      <BlockMath math="y' = -\frac{1}{4}x^{-3/2} + \frac{1}{2}" />
    </MathContainer>

    So, the first derivative is $$\frac{1}{2} - \frac{1}{4}x^{-3/2}$$.

Visible text: 1. For this problem, the easiest way is to simplify the function before differentiating it.

 **Step** : Split the Fraction

 We can break the fraction into two separate parts to make it easier.

 
 

 **Step** : Simplify Each Term

 Convert the square root to the exponent and use exponent properties to simplify each term.

 <MathContainer>
 
 

 
 

 
 

 </MathContainer>

 **Step** : Apply the Power Rule

 Once the function is simplified, we can directly differentiate it term by term.

 <MathContainer>
 <BlockMath math="y' = \frac{1}{2}\left(-\frac{1}{2}x^{-1/2 - 1}\right) + \frac{1}{2}(1)" />
 <BlockMath math="y' = -\frac{1}{4}x^{-3/2} + \frac{1}{2}" />
 </MathContainer>

 So, the first derivative is .