Derivative of x/3

We now understand the derivative of x/3. It is commonly represented as d/dx (x/3) or (x/3)', and its value is 1/3.

The function x/3 has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below: Linear Function: (f(x) = x/3).

Constant Rule: Rule for differentiating constant multiples of x.

Constant Function: The derivative of a constant is 0.

The derivative of x/3 can be denoted as d/dx (x/3) or (x/3)'.

The formula we use to differentiate x/3 is: d/dx (x/3) = 1/3 The formula applies to all x.

We can derive the derivative of x/3 using proofs. To show this, we will use the rule of differentiation for constant multiples.

There are several methods we use to prove this, such as: By First Principle Using Constant Rule Using Power Rule

We will now demonstrate that the differentiation of x/3 results in 1/3 using the above-mentioned methods: By First Principle The derivative of x/3 can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of x/3 using the first principle, we will consider f(x) = x/3. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = x/3, we write f(x + h) = (x + h)/3.

Substituting these into equation (1), f'(x) = limₕ→₀ [(x + h)/3 - x/3] / h = limₕ→₀ [h/3] / h = limₕ→₀ 1/3 Thus, f'(x) = 1/3. Hence, proved.

Using Constant Rule To prove the differentiation of x/3 using the constant rule, We use the formula: d/dx (c·x) = c Here, c = 1/3 Therefore, d/dx (x/3) = 1/3.

Using Power Rule We will now prove the derivative of x/3 using the power rule.

The step-by-step process is demonstrated below: Consider f(x) = x/3 = (1/3)x.

Using the power rule formula: d/dx [x^n] = n·x^(n-1) Here, n = 1. d/dx (x/3) = d/dx [(1/3)x] = (1/3)·d/dx [x^1] = (1/3)·1·x^(1-1) = 1/3

Thus, the derivative of x/3 is 1/3.

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 x/3. For the first derivative of a function, we write f′(x), 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′′(x). Similarly, the third derivative, f′′′(x) is the result of the second derivative and this pattern continues.

For the nth Derivative of x/3, we generally use f^(n)(x) for the nth derivative of a function f(x) which tells us the change in the rate of change.

For linear functions like x/3, higher-order derivatives will be zero beyond the first derivative.

For any value of x, the derivative of x/3 is always 1/3.

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

• Constant Rule: A rule stating that the derivative of a constant multiple of x is the constant itself.

• Linear Function: A function of the form f(x) = mx + b, here m = 1/3, b = 0.

• First Derivative: It is the initial result of a function, which gives us the rate of change of a specific function.

• Second Derivative: The derivative of the first derivative, which indicates the rate of change of the rate of change. For linear functions, this is zero.