Derivative of t/2

We now understand the derivative of t/2. It is commonly represented as d/dt (t/2) or (t/2)', and its value is 1/2. The function t/2 has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below:

Linear Function: A function of the form f(t) = at + b, where a and b are constants.

Constant Rule: The derivative of a constant is zero.

Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative of the function.

The derivative of t/2 can be denoted as d/dt (t/2) or (t/2)'.

The formula we use to differentiate t/2 is: d/dt (t/2) = 1/2 (or) (t/2)' = 1/2

The formula applies to all t.

We can derive the derivative of t/2 using proofs. To show this, we will use basic differentiation rules. There are several methods we use to prove this, such as:

1. By First Principle
2. Using Constant Multiple Rule
3. By First Principle

The derivative of t/2 can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of t/2 using the first principle, we will consider f(t) = t/2. Its derivative can be expressed as the following limit.

f'(t) = limₕ→₀ [f(t + h) - f(t)] / h … (1)

Given that f(t) = t/2, we write f(t + h) = (t + h)/2.

Substituting these into equation (1),

f'(t) = limₕ→₀ [(t + h)/2 - t/2] / h = limₕ→₀ [h/2] / h = limₕ→₀ 1/2 f'(t) = 1/2 Hence, proved.

Using Constant Multiple Rule

To prove the differentiation of t/2 using the constant multiple rule, We use the formula: If f(t) = t, then d/dt (f(t)) = 1. So, for f(t) = t/2,

we have: f'(t) = (1/2) * d/dt (t) f'(t) = (1/2) * 1 f'(t) = 1/2

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky.

To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like t/2.

For the first derivative of a function, we write f′(t), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, which is denoted using f′′(t). Similarly, the third derivative, f′′′(t) is the result of the second derivative and this pattern continues.

For the nth Derivative of t/2, we generally use fⁿ(t) for the nth derivative of a function f(t), which tells us the change in the rate of change. (continuing for higher-order derivatives).

When the t is 0, the derivative of t/2 is 1/2.

• Derivative: The derivative of a function indicates how the given function changes in response to a slight change in the variable.

• Linear Function: A function of the form at+b, where a and b are constants.

• Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative of the function.

• Product Rule: A rule for differentiating products of two functions.

• Quotient Rule: A rule for differentiating quotients of two functions.