Derivative of x/7

We now understand the derivative of x/7. It is commonly represented as d/dx (x/7) or (x/7)', and its value is 1/7. The function x/7 has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Linear Function: x/7 is a simple linear function.

Constant Rule: The derivative of a constant times a function.

The derivative of x/7 can be denoted as d/dx (x/7) or (x/7)'. The formula we use to differentiate x/7 is: d/dx (x/7) = 1/7

We can derive the derivative of x/7 using proofs. To show this, we will use basic rules of differentiation. There are several methods we use to prove this, such as:

1. By First Principle
2. Using Constant Multiplication Rule

By First Principle

The derivative of x/7 can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of x/7 using the first principle, we will consider f(x) = x/7. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h

Given that f(x) = x/7,

we write f(x + h) = (x + h)/7.

Substituting these into the equation, f'(x) = limₕ→₀ [(x + h)/7 - x/7] / h = limₕ→₀ [h/7] / h = limₕ→₀ 1/7 f'(x) = 1/7

Hence, proved.

Using Constant Multiplication Rule

To prove the differentiation of x/7 using the constant multiplication rule: Consider f(x) = x/7 We use the formula d/dx (c * f(x)) = c * d/dx (f(x)), where c is a constant.

Here, c = 1/7 and f(x) = x.

Thus, d/dx (x/7) = 1/7 * d/dx (x) = 1/7 * 1 = 1/7

Hence, the derivative is 1/7.

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 for more complex functions, but for x/7, it is straightforward.

The first derivative is a constant, so all higher-order derivatives will be zero. 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/7, we generally use fⁿ(x). Since the first derivative is constant, all higher-order derivatives (second, third, etc.) are zero.

Since x/7 is a linear function with a constant slope, there are no points of discontinuity or undefined behavior. Therefore, there are no special cases where the derivative changes behavior within its domain.

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

• Constant Rule: The rule stating that the derivative of a constant times a function is the constant times the derivative of the function.

• First Derivative: The initial result of differentiating a function, providing the rate of change.

• Linear Function: A function of the form y = mx + b, where m and b are constants.

• Higher-Order Derivatives: Derivatives obtained by differentiating a function multiple times, providing insights into the rate of change of the rate of change.