Chegg 6 Calculate The Following Derivative D Dx E-T3 Dt

Introduction

Calculating derivatives is a fundamental skill in calculus. The derivative of a function measures how the function's value changes as its input changes. For the function e-t3, we'll find its derivative with respect to t.

This calculation is important in physics, engineering, and economics where exponential functions are common. The derivative of e-t3 helps analyze growth rates, decay rates, and other rate-related problems.

Derivative Rules

Before calculating the derivative of e-t3, let's review the key rules we'll use:

1. Power Rule: The derivative of x^n is n*x^(n-1).
2. Exponential Rule: The derivative of e^x is e^x.
3. Chain Rule: Used when differentiating composite functions.

For e-t3, we'll apply the exponential rule and the chain rule since the exponent is a function of t.

Step-by-Step Solution

Let's calculate the derivative of e-t3 step by step:

1. Identify the function: f(t) = e-t3
2. Apply the chain rule: d/dt [e-u] = e-u * du/dt, where u = t3
3. Differentiate the exponent: du/dt = d/dt [t3] = 3t2
4. Combine the results: d/dt [e-t3] = e-t3 * 3t2

This result shows that the derivative of e-t3 is a product of the original function and the derivative of its exponent.

Worked Example

Let's calculate the derivative of e-t3 at t = 1:

1. First, find the derivative: d/dt [e-t3] = 3t2 * e-t3
2. Substitute t = 1: d/dt [e-13] = 3(1)2 * e-13 = 3 * e-13
3. Calculate the numerical value: e ≈ 2.71828, so e-13 ≈ 0.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000