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129 LearnersLast updated on September 15, 2025
Derivative of sqrt of x
The derivative of sqrt(x), which is 1/(2sqrt(x)), is used as a tool to understand how the square root function changes with respect to a small change in x. Derivatives are crucial in many real-life applications, such as determining rates of change. We will now delve into the details of the derivative of sqrt(x).
What is the Derivative of sqrt of x?
We now understand the derivative of sqrt(x). It is commonly represented as d/dx (sqrt(x)) or (sqrt(x))', and its value is 1/(2sqrt(x)). The function sqrt(x) has a well-defined derivative, indicating it is differentiable for x > 0.
The key concepts are mentioned below:
Square Root Function: sqrt(x) is the square root of x.
Power Rule: A rule for differentiating functions of the form x^n.
Reciprocal Function: For a function f(x), its reciprocal is 1/f(x).
Derivative of sqrt of x Formula
The derivative of sqrt(x) can be denoted as d/dx (sqrt(x)) or (sqrt(x))'.
The formula we use to differentiate sqrt(x) is: d/dx (sqrt(x)) = 1/(2sqrt(x))
The formula applies to all x where x > 0.
Proofs of the Derivative of sqrt of x
We can derive the derivative of sqrt(x) using various methods. Let's explore some of these methods:
Using the Power Rule
The function sqrt(x) can be rewritten as x(1/2). Applying the power rule, which states d/dx (x^n) = n*x^(n-1), we have: d/dx (x(1/2)) = (1/2)*x(-1/2) = 1/(2sqrt(x)).
By First Principle
The derivative of sqrt(x) can also be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. Consider f(x) = sqrt(x). Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h = limₕ→₀ [sqrt(x + h) - sqrt(x)] / h
To rationalize the numerator, multiply and divide by the conjugate: = limₕ→₀ [(sqrt(x + h) - sqrt(x)) * (sqrt(x + h) + sqrt(x))] / [h * (sqrt(x + h) + sqrt(x))] = limₕ→₀ [h] / [h * (sqrt(x + h) + sqrt(x))] = limₕ→₀ 1 / (sqrt(x + h) + sqrt(x))
As h approaches 0, this becomes: = 1/(2sqrt(x)). Hence, proved.
Higher-Order Derivatives of sqrt of x
When a function is differentiated multiple times, the resulting derivatives are called higher-order derivatives. Higher-order derivatives can be complex, especially for functions like sqrt(x).
For the first derivative of sqrt(x), we denote f′(x) = 1/(2sqrt(x)), which indicates the rate of change of the function. The second derivative is derived from the first derivative and provides information about the concavity of the function.
For sqrt(x), the second derivative is negative, indicating the function is concave down. Higher-order derivatives can be denoted as f^(n)(x) for the nth derivative and offer deeper insights into the behavior of the function.
Special Cases:
When x approaches 0 from the positive side, the derivative 1/(2sqrt(x)) becomes very large, reflecting the steepness of the curve near the origin. When x is 1, the derivative of sqrt(x) = 1/(2sqrt(1)) = 1/2.
Common Mistakes and How to Avoid Them in Derivatives of sqrt of x
Students frequently make mistakes when differentiating sqrt(x). These mistakes can be resolved by understanding the correct methods. Here are a few common mistakes and ways to solve them:
Mistake 1
Not Simplifying the Expression
Students may forget to simplify the expression before differentiating, which can lead to incorrect results. For example, not recognizing that sqrt(x) is x^(1/2) can cause errors. Always simplify expressions where possible.
Mistake 2
Ignoring the Domain of sqrt(x)
Students might overlook that sqrt(x) is only defined for x ≥ 0. While differentiating, ensure you consider the domain to avoid misinterpretations.
Mistake 3
Incorrect Use of the Power Rule
While differentiating functions like sqrt(x), students might misapply the power rule. For example, incorrectly writing d/dx (sqrt(x)) as 1/sqrt(x) is a common mistake. Remember that sqrt(x) is x^(1/2) and apply the power rule correctly.
Mistake 4
Forgetting to Rationalize the Denominator
When using the first principle, students might forget to rationalize the denominator, leading to complex expressions. Always remember to rationalize to simplify the process.
Mistake 5
Not Considering Higher-Order Derivatives
Students often stop at the first derivative and forget to explore higher-order derivatives. These derivatives provide valuable insights into the function's behavior, so consider them where applicable.
Examples Using the Derivative of sqrt of x
Problem 1
Calculate the derivative of (sqrt(x) * x^2)
Here, we have f(x) = sqrt(x) * x2. Using the product rule, f'(x) = u′v + uv′ In the given equation, u = sqrt(x) and v = x2.
Let’s differentiate each term, u′ = d/dx (sqrt(x)) = 1/(2sqrt(x)) v′ = d/dx (x2) = 2x
Substituting into the equation, f'(x) = (1/(2sqrt(x))) * x2 + sqrt(x) * 2x = x(3/2)/2 + 2x(3/2) = (5/2)x(3/2)
Thus, the derivative of the specified function is (5/2)x(3/2).
Explanation
We find the derivative of the given function by dividing it into two parts, then applying the product rule to find the final result.
Problem 2
A water tank is being filled, and the height of the water is modeled by h(x) = sqrt(x) meters, where x is the time in minutes. Find the rate of change of the height at x = 4 minutes.
We have h(x) = sqrt(x) (height of water)...(1)
Now, we will differentiate the equation (1). Take the derivative of sqrt(x): dh/dx = 1/(2sqrt(x))
Given x = 4 (substitute this into the derivative): dh/dx = 1/(2sqrt(4)) = 1/(2*2) = 1/4
Hence, the rate of change of the height at x = 4 minutes is 1/4 meters per minute.
Explanation
We find the rate of change of the height at x= 4 minutes, indicating the height increases by 1/4 meter per minute at that time.
Problem 3
Derive the second derivative of the function y = sqrt(x).
The first step is to find the first derivative, dy/dx = 1/(2sqrt(x))...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [1/(2sqrt(x))] = d/dx [(1/2) * x(-1/2)] = (1/2) * (-1/2) * x(-3/2) = -1/(4x(3/2))
Therefore, the second derivative of the function y = sqrt(x) is -1/(4x(3/2)).
Explanation
We differentiate the first derivative using the power rule to find the second derivative, providing insights into the concavity of the function.
Problem 4
Prove: d/dx (x * sqrt(x)) = 3/2 * sqrt(x).
Let's start using the product rule: Consider y = x * sqrt(x) = x * x(1/2) = x(3/2)
To differentiate, apply the power rule: dy/dx = (3/2) * x(1/2) = 3/2 * sqrt(x) Hence proved.
Explanation
In this step-by-step process, we rewrite the function as x(3/2) and apply the power rule to differentiate, leading to the desired result.
Problem 5
Solve: d/dx (sqrt(x)/x)
To differentiate the function, we use the quotient rule:
d/dx (sqrt(x)/x) = (d/dx (sqrt(x)) * x - sqrt(x) * d/dx(x)) / x² = (1/(2sqrt(x)) * x - sqrt(x) * 1) / x² = (x/(2sqrt(x)) - sqrt(x)) / x² = (x - 2x) / (2x(3/2)) = -1/(2x(1/2))
Therefore, d/dx (sqrt(x)/x) = -1/(2sqrt(x)).
Explanation
We differentiate the given function using the quotient rule, simplifying the equation to obtain the final result.
FAQs on the Derivative of sqrt of x
1.Find the derivative of sqrt(x).
2.Can we use the derivative of sqrt(x) in real life?
3.Is it possible to take the derivative of sqrt(x) at x = 0?
4.What rule is used to differentiate sqrt(x)/x?
5.Are the derivatives of sqrt(x) and sqrt(x^2) the same?
Important Glossaries for the Derivative of sqrt of x
- Derivative: The derivative of a function indicates how the function changes with respect to a change in x.
- Square Root Function: A function that returns the principal square root of x, denoted as sqrt(x).
- Power Rule: A basic rule in calculus used to differentiate functions of the form x^n.
- Quotient Rule: A rule used to differentiate functions represented as one function divided by another.
- Reciprocal Function: A function that represents the inverse of another function, typically represented as 1/f(x).
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
