Derivative of the Square Root of x

Finding the Derivative of the Square Root of x

In the realm of calculus, derivatives play a crucial role in understanding the rate of change of functions. One common function we encounter is the square root of x, denoted as √x. Finding its derivative is a fundamental concept, and in this blog post, we'll delve into the process using the power rule.

The Power Rule

The power rule states that the derivative of x raised to the power of n (xⁿ) is equal to n times x raised to the power of n-1 (nx^n-1). Mathematically:

d/dx (xⁿ) = nx^n-1

Applying the Power Rule to √x

First, we need to express √x in terms of a power of x. Recall that the square root of x is equivalent to x raised to the power of 1/2 (x^1/2). Now, we can apply the power rule:

d/dx (√x) = d/dx (x^1/2) = (1/2)x^(1/2)-1

Simplifying the exponent, we get:

d/dx (√x) = (1/2)x^-1/2

Finally, we can rewrite the expression with a positive exponent by moving x^-1/2 to the denominator:

d/dx (√x) = 1 / (2√x)

Example

Let's find the derivative of √x at x = 4:

d/dx (√x) at x = 4 = 1 / (2√4) = 1 / (2 * 2) = 1/4

Conclusion

Using the power rule, we successfully derived the derivative of the square root of x, which is 1 / (2√x). This process demonstrates the power and simplicity of the power rule in calculus, allowing us to find derivatives of functions with ease. Understanding the derivative of √x is essential for various applications in calculus, such as finding the slope of a tangent line or analyzing the rate of change of a function involving square roots.