DIFFERENTIATE 2^t to SOLVE a DIFFERENTIAL Equation! 🧠

How do you take the derivative of an exponential function with a base other than e, and how does it verify a differential equation?

Deriving p(t) = p_0 * (2^t) Step-by-Step:
• Derivative Rule for Base b: d/dt [b^t] = ln(b) * (b^t).
• Computing the Derivative: p'(t) = dp/dt = ln(2) * p_0 * (2^t).
• Verifying the Differential Equation: Since p(t) = p_0 * (2^t), substituting gives dp/dt = ln(2) * p(t) for all t! Therefore, p(t) solves the differential equation dp/dt = ln(2)*p.

Notation Reminder:
Whether you write prime notation p'(t) or Leibniz notation dp/dt, both describe the exact same instantaneous rate of change!

Verifying that the derivative of a function equals a constant times the function itself is the cornerstone of exponential growth in continuous differential equations!

📚 Resources & Full Lectures:
• Watch the full 69-minute Lecture 1:
• Read my “Visual Linear Algebra Online” chapter (“Points, Coordinates, Skew Coordinates, and Graphs”):

#DifferentialEquations #Calculus #ExponentialGrowth #Derivatives #MathHelp #LeibnizNotation #CollegeMath #visualmath

Pinned Comment: Quick calculus check: Do you remember why d/dt [2^t] gets that extra factor of ln(2)? 🧠

Since e^t is its own derivative, any other base b requires multiplying by ln(b)! That’s why p(t) = p_0 * (2^t) solves the differential equation dp/dt = ln(2) * p!

WATCH THE FULL LECTURE 1 HERE:
EXPLORE VISUAL LINEAR ALGEBRA ONLINE:

Do you prefer prime notation p'(t) or Leibniz notation dp/dt when doing differential equations? Let me know below! 👇

📖 Differential Equations, by Blanchard, Devaney, and Hall:

Links and resources
===============================
🔴 Subscribe to Bill Kinney Math:
🔴 Subscribe to my Math Blog, Infinity is Really Big:
🔴 Follow me on LinkedIn:
🔴 Follow me on Twitter:
🔴 Follow me on Instagram:
🔴 You can support me by buying “Infinite Powers, How Calculus Reveals the Secrets of the Universe”, by Steven Strogatz, or anything else you want to buy, starting from this link:
🔴 Check out my artist son Tyler Kinney’s website:
🔴 Desiring God website:

AMAZON ASSOCIATE
As an Amazon Associate I earn from qualifying purchases.

source