Derivative of Algebraic Function

Applying the Derivative Rules

The derivative rules apply to many algebraic forms. For a polynomial, rational function, or expression containing a radical, first identify the structure and then choose the matching rule. The examples below make that decision explicit.

Using the Properties of Derivatives

Each example uses a different algebraic form and explains why its method works. Before differentiating, rewrite a quotient or a root as a power, because the power rule then applies directly to every term.

Differentiating a Polynomial

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

Solution:

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

Differentiating a Radical Expression

For $x>0$, find the first derivative of $y = x\sqrt{x^3}$.

Solution:

There are two ways to solve this.

Method 1: Using the Product Rule

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

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

Method 2: Simplifying First

We can simplify the function before differentiating it.

Both methods produce $y'=\frac{5}{2}x^{3/2}$. Simplifying first takes fewer steps. The product rule differentiates the two factors separately, so it provides an independent check of the same derivative. As a real-valued function, the original expression has domain $x \ge 0$. The differentiation above is carried out on its interior, $x>0$.

Differentiating a Rational Function

On its domain $x\ne0$, find the first derivative of $y = 2\left(\frac{x^2+2}{x}\right)$.

Solution:

We use the quotient rule. Let $u(x) = x^2+2$ and $v(x) = x$. Then $u'(x) = 2x$ and $v'(x) = 1$.

The restriction $x\ne0$ remains part of the derivative's domain.

Exercises

Each problem asks for the derivative of an algebraic expression. Simplify the expression where possible, then apply the power rule.

1. For $x>0$, find the first derivative of $y = \frac{\sqrt{x} + x^2}{2x}$.

Answer Key

1. Rewrite the function as a sum of powers. The power rule can then be applied to each term.

   Step 1: Split the Fraction

   Split the two terms in the numerator over the common denominator:

   $$y = \frac{\sqrt{x}}{2x} + \frac{x^2}{2x}$$

   Step 2: Simplify Each Term

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

   $$y = \frac{x^{1/2}}{2x^1} + \frac{x^2}{2x^1}$$

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

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

   Step 3: Apply the Power Rule

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

   $$y' = \frac{1}{2}\left(-\frac{1}{2}x^{-1/2 - 1}\right) + \frac{1}{2}(1)$$

   $$y' = -\frac{1}{4}x^{-3/2} + \frac{1}{2}$$

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