D/dx(2x^3 + 4x^2 - 5x) = 6x^2 + 8x - 5

Definition

This expression represents the derivative of the polynomial function $$f(x) = 2x^3 + 4x^2 - 5x$$ with respect to the variable x. The process of differentiation involves applying rules to find the rate at which the function changes, and in this case, the result indicates how steeply the function is increasing or decreasing at any point on its curve.

5 Must Know Facts For Your Next Test

1. The derivative is calculated using the power rule, which states that for any term $$ax^n$$, the derivative is $$na x^{n-1}$$.
2. In the expression, each term is differentiated individually: for $$2x^3$$ it becomes $$6x^2$$, for $$4x^2$$ it becomes $$8x$$, and for $$-5x$$ it results in $$-5$$.
3. Constant terms have a derivative of zero, which is why the constant component of the polynomial, if any, does not appear in the final result.
4. This process helps find critical points where the function's slope equals zero, indicating potential maximum or minimum values.
5. The final result, $$6x^2 + 8x - 5$$, is another polynomial function that provides insights into the original function's behavior.

Review Questions

• How does applying the power rule allow us to differentiate each term in the polynomial function?
  • The power rule simplifies differentiation by providing a straightforward method for finding the derivative of each term in a polynomial. For each term of the form $$ax^n$$, we multiply by n and decrease the exponent by one. In this example, applying this rule gives us derivatives of 6x² from 2x³, 8x from 4x², and -5 from -5x. This approach demonstrates how easy it can be to differentiate polynomial expressions quickly.
• What role does understanding derivatives play in analyzing polynomial functions like $$f(x) = 2x^3 + 4x^2 - 5x$$?
  • Understanding derivatives is crucial for analyzing polynomial functions because they provide information about the function's behavior. By determining where the derivative equals zero, we can locate critical points that may represent local maxima or minima. Additionally, the sign of the derivative indicates whether the function is increasing or decreasing at certain intervals. Thus, analyzing the derivative helps us sketch graphs and understand key features of polynomial functions.
• Evaluate how knowing the derivative affects our interpretation of real-world problems modeled by polynomial functions.
  • Knowing the derivative allows us to interpret real-world problems by understanding rates of change and optimizing situations modeled by polynomial functions. For instance, in physics, it might represent velocity or acceleration based on position functions, helping predict motion. In economics, derivatives can show how costs change with production levels. Ultimately, interpreting derivatives enables decision-making based on how systems respond to changes in variables.

Related terms

Derivative: A measure of how a function's output value changes as its input value changes, representing the slope of the tangent line to the function at a given point.

Polynomial Function: A mathematical expression involving a sum of powers in one or more variables multiplied by coefficients, such as $$f(x) = ax^n + bx^{n-1} + ... + k$$.

Rate of Change: The ratio that describes how much one quantity changes in relation to another quantity, often represented by the derivative in calculus.