Derivative of te^t

We now understand the derivative of te^t. It is commonly represented as d/dt (te^t) or (te^t)', and its value is e^t + te^t.

The function te^t has a clearly defined derivative, indicating it is differentiable within its domain. Key concepts include: 

Exponential Function: (e^t is the exponential function with base e). 

Product Rule: Used for differentiating te^t (since it consists of t multiplied by e^t).

The derivative of te^t can be denoted as d/dt (te^t) or (te^t)'. The formula we use to differentiate te^t is: d/dt (te^t) = e^t + te^t This formula applies to all t in the real number domain.

We can derive the derivative of te^t using proofs. To show this, we will use differentiation rules. Several methods can prove this: -

1. By First Principle 
2. Using Product Rule

Let's demonstrate that differentiating te^t results in e^t + te^t using these methods:

Using First Principle

The derivative of te^t can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of te^t using the first principle, consider f(t) = te^t. Its derivative can be expressed as the following limit: f'(t) = limₕ→₀ [f(t + h) - f(t)] / h

Given f(t) = te^t, we write f(t + h) = (t + h)e^(t + h).

Substituting these into the equation, f'(t) = limₕ→₀ [(t + h)e^(t + h) - te^t] / h = limₕ→₀ [te^(t + h) + he^(t + h) - te^t] / h = limₕ→₀ [te^(t + h) - te^t + he^(t + h)] / h = limₕ→₀ [t(e^(t + h) - e^t) + he^(t + h)] / h

Using the limit property e^(t + h) = e^t * e^h and the fact limₕ→₀ (e^h - 1)/h = 1, f'(t) = te^t * 1 + e^t = e^t + te^t

Using Product Rule

To prove the differentiation of te^t using the product rule, Consider f(t) = t and g(t) = e^t so that te^t = f(t)g(t).

By the product rule: d/dt [f(t)g(t)] = f'(t)g(t) + f(t)g'(t) f'(t) = 1 (derivative of t) g'(t) = e^t (derivative of e^t)

Substituting into the product rule, d/dt (te^t) = 1(e^t) + t(e^t) = e^t + te^t

When a function is differentiated multiple times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives offer deeper insights into the behavior of functions.

For instance, consider a vehicle where the velocity changes (first derivative), and the rate at which the velocity changes (second derivative) also varies. Higher-order derivatives help us understand functions like te^t more clearly.

For the first derivative of a function, we denote it as f′(t). The second derivative, f′′(t), is obtained from the first derivative. Similarly, the third derivative, f′′′(t), results from the second derivative, continuing this pattern.

For the nth Derivative of te^t, we generally use f n(t) for the nth derivative of a function f(t), which tells us the change in the rate of change.

- When t is 0, the derivative of te^t = e^0 + 0 * e^0, which is 1. - As t approaches -∞, the derivative approaches 0 since e^t diminishes.

• Derivative: Indicates how a function changes in response to a slight change in its input variable. 

• Exponential Function: A function of the form e^t, where e is the base of natural logarithms. 

• Product Rule: A differentiation rule used to differentiate the product of two functions. 

• Higher-Order Derivative: Derivatives obtained by differentiating a function multiple times. 

• First Principle: A fundamental method of finding derivatives using the limit of the difference quotient.