Calculate The Following Derivative D Dx Z X4 X2

Calculating the derivative of z = x⁴ + x² with respect to x is a fundamental calculus operation. This guide explains the process, provides a calculator, and includes a step-by-step example.

How to Calculate the Derivative

The derivative of a function measures how the function's output changes as its input changes. For z = x⁴ + x², we'll find dz/dx by applying the basic rules of differentiation.

Key Concept: The derivative of a sum is the sum of the derivatives. The derivative of xⁿ is n*x^(n-1).

Basic Rules Used

Power Rule: d/dx (xⁿ) = n*x^(n-1)
Sum Rule: d/dx (f(x) + g(x)) = f'(x) + g'(x)
Constant Rule: d/dx (c) = 0 (where c is a constant)

Step-by-Step Calculation

Identify the function: z = x⁴ + x²
Apply the power rule to each term:
- d/dx (x⁴) = 4x³
- d/dx (x²) = 2x
Combine the results using the sum rule: dz/dx = 4x³ + 2x

Final Derivative: dz/dx = 4x³ + 2x

The Derivative Formula

The general formula for the derivative of z = x⁴ + x² is:

dz/dx = d/dx (x⁴) + d/dx (x²) = 4x³ + 2x

This formula shows that the derivative of each term is calculated separately and then combined.

Worked Example

Let's calculate dz/dx when x = 3:

Start with the derivative: dz/dx = 4x³ + 2x
Substitute x = 3: dz/dx = 4(3)³ + 2(3)
Calculate each term:
- 4(3)³ = 4 × 27 = 108
- 2(3) = 6
Add the results: 108 + 6 = 114

Result

When x = 3, dz/dx = 114

Frequently Asked Questions

What is the derivative of x⁴ + x²?: The derivative is 4x³ + 2x, calculated using the power rule and sum rule.
Can I use this calculator for other functions?: This calculator is specifically for z = x⁴ + x². For other functions, you would need a different calculator.
What if I want to find the derivative at a specific point?: Enter the x value in the calculator and it will compute the derivative at that point.
Is the derivative always a straight line?: No, the derivative represents the slope of the tangent line at any point, which can change as x changes.
Where can I learn more about calculus?: We recommend checking out our Calculus Fundamentals guide and Differential Calculus calculator.