R'(t) = racddt(3t^3 - 5t^2 + 2t + 10) = 9t^2 - 10t + 2. - RTA

Understanding the Context

What Does the Notation Mean?

We start with a polynomial function:
R(t) = 3t³ - 5t² + 2t + 10
The notation R’(t) = d/dt (3t³ - 5t² + 2t + 10) signifies the derivative of R(t) with respect to t. Derivatives measure the instantaneous rate of change or slope of a function at any point.

Using the basic rules of differentiation:

• The derivative of tⁿ is n tⁿ⁻¹
  
• Constants vanish (derivative of 10 is 0)
  
• Derivatives act linearly over sums

Key Insights

So applying these rules term by term:

• d/dt(3t³) = 3 × 3t² = 9t²
  
• d/dt(-5t²) = -5 × 2t = -10t
  
• d/dt(2t) = 2 × 1 = 2
  
• d/dt(10) = 0

Adding them together:
R’(t) = 9t² - 10t + 2

Why Is This Derivative Important?

Final Thoughts

Differentiating polynomials like this reveals crucial information:

• Slope at any point: R’(t) gives the slope of the original function R(t) at any value of t, indicating whether the function is increasing, decreasing, or flat.
  
• Graph behavior: Helps identify critical points (where slope = 0) used in optimization and analysis of maxima/minima.
  
• Real-world applications: In physics, derivatives represent velocity (derivative of position) or acceleration (derivative of velocity); in economics, marginal cost or revenue rely on such rates of change.

Visual Insight: Graph of R(t) and R’(t)

Imagine the cubic-shaped curve of R(t) = 3t³ - 5t² + 2t + 10 — steep growth for large positive t, with bends controlled by coefficients. Near specific t-values, the derivative R’(t) = 9t² - 10t + 2 quantifies how sharply R(t) rises or falls, helping sketch tangent lines with precise slopes at each point.