D/dx(3x^2 + 5) = 6x + 0

Definition

This expression represents the derivative of the function 3x^2 + 5 with respect to x, showing how the function changes as x changes. It highlights two key concepts: the power rule for differentiating polynomial terms and the constant rule which states that the derivative of a constant is zero. Understanding this expression helps in grasping how to find slopes of curves and analyze the behavior of functions.

5 Must Know Facts For Your Next Test

1. The derivative of 3x^2 is found using the power rule, which tells us to multiply by 2 (the exponent) and then subtract 1 from the exponent.
2. The constant term, 5, has a derivative of zero because it does not change regardless of the value of x.
3. When combined, d/dx(3x^2 + 5) simplifies to 6x + 0, indicating that the slope of the function changes linearly with respect to x.
4. This derivative can be evaluated at any point x to find the instantaneous rate of change or slope at that specific value.
5. Understanding this expression is foundational for solving more complex problems involving rates of change in calculus.

Review Questions

• How does applying the power rule help in differentiating polynomial expressions like 3x^2?
  • The power rule simplifies finding derivatives for polynomial terms by allowing us to multiply by the exponent and reduce it by one. In this case, for 3x^2, applying the power rule results in 2 times 3, giving us 6 as a coefficient for x raised to the power of 1. This makes differentiation straightforward and efficient for any polynomial.
• What does it mean when we say that d/dx(5) = 0 in relation to understanding function behavior?
  • When we say d/dx(5) = 0, it indicates that a constant does not contribute to changes in function behavior since it remains unchanged regardless of x. This helps in identifying parts of a function that influence its overall shape and slope. The presence of constants like this suggests flatness in those regions on a graph, emphasizing that only variable terms affect how steep or shallow a curve can be.
• Evaluate how understanding derivatives like d/dx(3x^2 + 5) impacts solving real-world problems involving motion or growth.
  • Understanding derivatives such as d/dx(3x^2 + 5) allows us to model real-world situations where rates of change are critical, such as calculating velocity in motion or growth rates in populations. By applying concepts like the power and constant rules, we can derive functions that accurately represent these phenomena. This analysis helps predict future behaviors based on current trends, enabling informed decision-making in various fields such as physics, biology, and economics.

Related terms

Derivative: The derivative of a function measures how the function's output changes as its input changes, representing the slope of the tangent line to the curve at any point.

Power Rule: A method used to find the derivative of a term in the form of ax^n, where the derivative is obtained by multiplying by the exponent and decreasing the exponent by one.

Constant Rule: A principle stating that the derivative of any constant value is zero, reflecting that constants do not change as their variable inputs change.