D/dx(3x^5) = 15x^4

Definition

This expression represents the derivative of the function 3x^5 with respect to x, which results in 15x^4. Derivatives measure how a function changes as its input changes, and in this case, it highlights the application of both the power rule and the constant rule in calculus. Understanding this term is crucial for analyzing how polynomial functions behave and how their rates of change can be calculated efficiently.

5 Must Know Facts For Your Next Test

1. When applying the power rule, you bring down the exponent (5 in this case) and multiply it by the coefficient (3), resulting in 15.
2. After multiplying by the coefficient, you decrease the exponent by one, changing it from 5 to 4.
3. This process illustrates that polynomials can be differentiated efficiently using these rules without needing to expand them.
4. Understanding derivatives like d/dx(3x^5) = 15x^4 is fundamental for studying rates of change and optimization problems in calculus.
5. Using these rules helps simplify complex differentiation tasks into straightforward calculations.

Review Questions

• How do you apply the power rule to find the derivative of a polynomial function like 3x^5?
  • To apply the power rule to find the derivative of 3x^5, you first identify the exponent, which is 5. According to the power rule, you multiply the coefficient (3) by this exponent, giving you 15. Then, you decrease the exponent by one, changing it from 5 to 4. So, the derivative d/dx(3x^5) results in 15x^4.
• What role does the constant rule play when differentiating a term like 3x^5?
  • The constant rule is essential when differentiating terms that include coefficients. In the case of 3x^5, this rule indicates that when you take the derivative, you treat the constant (3) as a multiplier. Therefore, when applying both the constant rule and power rule together, you multiply 3 by the derivative obtained from applying the power rule to x^5, ultimately simplifying the process to d/dx(3x^5) = 15x^4.
• Evaluate how understanding d/dx(3x^5) = 15x^4 can help in solving real-world problems involving rates of change.
  • Understanding d/dx(3x^5) = 15x^4 provides valuable insight into solving real-world problems involving rates of change because it illustrates how to calculate derivatives effectively. In many applications, such as physics and economics, knowing how a quantity changes with respect to another is crucial for optimizing processes or making predictions. By mastering this derivative computation, one can model situations where polynomial relationships exist and analyze their behavior over time or under varying conditions.

Related terms

Derivative: A derivative is a measure of how a function's output value changes as its input value changes, providing insight into the function's behavior at any given point.

Power Rule: The power rule states that the derivative of x raised to the power of n is n*x^(n-1), simplifying the process of differentiating polynomial functions.

Constant Rule: The constant rule states that the derivative of a constant multiplied by a function is simply the constant multiplied by the derivative of that function.