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Calculate The Following D Dx X Arctan X3 1

Reviewed by Calculator Editorial Team

This guide explains how to calculate the derivative of x arctan(x³ + 1) using calculus rules. We'll cover the formula, step-by-step process, and practical applications of this mathematical operation.

How to Calculate d/dx x arctan(x³ + 1)

Calculating the derivative of x arctan(x³ + 1) requires applying the product rule and chain rule of differentiation. The product rule states that the derivative of a product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first.

d/dx [f(x) * g(x)] = f'(x) * g(x) + f(x) * g'(x)

In our case, f(x) = x and g(x) = arctan(x³ + 1). We'll need to find f'(x) and g'(x) separately before applying the product rule.

The Formula

The derivative of x arctan(x³ + 1) is calculated using the following formula:

d/dx [x arctan(x³ + 1)] = arctan(x³ + 1) + x * (3x² / (1 + (x³ + 1)²))

This formula combines the product rule with the chain rule applied to the arctangent function. The derivative of arctan(u) is 1/(1 + u²) * du/dx.

Step-by-Step Calculation

  1. Identify f(x) = x and g(x) = arctan(x³ + 1)
  2. Find f'(x) = 1 (derivative of x)
  3. Find g'(x) using the chain rule:
    • Let u = x³ + 1
    • Then g(x) = arctan(u)
    • g'(x) = (1 / (1 + u²)) * du/dx = (1 / (1 + (x³ + 1)²)) * 3x²
  4. Apply the product rule:
    • d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
    • = 1 * arctan(x³ + 1) + x * (3x² / (1 + (x³ + 1)²))

Worked Example

Let's calculate the derivative at x = 1:

Step Calculation Result
1. Calculate x³ + 1 1³ + 1 = 1 + 1 2
2. Calculate arctan(2) arctan(2) ≈ 1.107 radians ≈1.107
3. Calculate denominator 1 + (2)² = 1 + 4 5
4. Calculate g'(1) (1/5) * 3(1)² = 3/5 0.6
5. Apply product rule 1.107 + 1 * 0.6 ≈1.707

The derivative at x = 1 is approximately 1.707.

Frequently Asked Questions

What is the derivative of arctan(x)?
The derivative of arctan(x) is 1/(1 + x²). This is a standard calculus result that we use in our calculation.
When would I need to calculate d/dx x arctan(x³ + 1)?
This derivative appears in physics problems involving inverse trigonometric functions, optimization problems with constraints, and certain types of differential equations.
Can I use this calculator for other similar functions?
Yes, the same approach can be applied to functions of the form x arctan(f(x)) by adjusting the inner function f(x) in the calculator.
What are the units for the result?
The result is dimensionless since it's a derivative. The units depend on the original function's units.