Learn how to find the derivative of sqrt(1+x^2). The derivative is d/dx[sqrt(1+x^2)].
Below is the graph of f(x) = sqrt(1+x^2) and its derivative f'(x). Notice how the original function and derivative relate to each other.
Original Function:
Derivative:
This derivative represents the rate of change of the original function. The calculation follows standard differentiation rules as shown in the step-by-step solution below.
The function sqrt(1+x^2) involves a square root, which can be rewritten as (1+x^2)^(1/2). The Power Rule is applicable here because we have a term raised to a power, specifically a fractional power.
💡 Why this works: By rewriting sqrt(1+x^2) as (1+x^2)^(1/2), we can now apply the Power Rule, which is designed to differentiate functions where a variable is raised to a constant power.
Using the Power Rule, we differentiate (1+x^2)^(1/2). First, we bring down the exponent (1/2) and decrease the exponent by one. This gives us (1/2) * (1+x^2)^(-1/2). We then differentiate the inside term (1+x^2) with respect to x, which gives 2x.
💡 Why this works: So, applying the Power Rule results in the expression (1/2) * (1+x^2)^(-1/2) * 2x, which simplifies to x / sqrt(1+x^2). This is the derivative of sqrt(1+x^2).
Simplifying the result, we arrive at x / sqrt(1+x^2). This is the final derivative expression for the function sqrt(1+x^2).
💡 Why this works: The simplified form, x / sqrt(1+x^2), is the correct derivative of the function sqrt(1+x^2). This can be verified through a second check using standard differentiation rules.
[Teaching explanation - to be filled]
[Application to this function - to be filled]
❌ Common Mistake:
Not simplifying the final answer
✅ Correct Approach:
Always simplify your derivative to its most reduced form
Double-check your work by comparing your steps with our solution. Use our calculator to verify each step of the differentiation process.
Calculate any derivative instantly with step-by-step solutions. Perfect for students, teachers, and anyone learning calculus.