Derivative of 9x

We now understand the derivative of 9x. It is commonly represented as d/dx (9x) or (9x)', and its value is 9.
 

The function 9x has a clearly defined derivative, indicating it is differentiable across all real numbers.

The key concepts are mentioned below: Linear Function: A function of the form f(x) = ax + b.

Constant Rule: The derivative of a constant is zero.

Coefficient Rule: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.

The derivative of 9x can be denoted as d/dx (9x) or (9x)'.
 

The formula we use to differentiate 9x is: d/dx (9x) = 9 The formula applies to all values of x.

We can derive the derivative of 9x using basic calculus rules. To show this, we will use the fundamental rules of differentiation.
 

There are several methods we use to prove this, such as:

By Definition Using Constant Rule Using Coefficient Rule We will now demonstrate that the differentiation of 9x results in 9 using the above-mentioned methods:

By Definition The derivative of 9x can be proved using the definition of the derivative, which expresses the derivative as the limit of the difference quotient. To find the derivative of 9x using the definition, we will consider f(x) = 9x.

Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = 9x, we write f(x + h) = 9(x + h).

Substituting these into equation (1), f'(x) = limₕ→₀ [9(x + h) - 9x] / h = limₕ→₀ [9x + 9h - 9x] / h = limₕ→₀ [9h] / h = limₕ→₀ 9 = 9 Hence, proved. Using Constant Rule To prove the differentiation of 9x using the constant rule, We use the formula: d/dx (c*f(x)) = c * f'(x) Here, c = 9 and f(x) = x Since the derivative of f(x) = x is 1, d/dx (9x) = 9 * 1 = 9

Using Coefficient Rule We will now prove the derivative of 9x using the coefficient rule.

The step-by-step process is demonstrated below: Given that, u(x) = x and c = 9

Using the coefficient rule formula: d/dx [c*u(x)] = c * u'(x) u'(x) = d/dx (x) = 1 Thus, d/dx (9x) = 9 * 1 = 9.

Hence, proved.

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 9x. 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 9x, we generally use fⁿ(x) for the nth derivative of a function f(x) which tells us the change in the rate of change (continuing for higher-order derivatives).

The function 9x is a linear function and does not have special cases of undefined points or asymptotes. For any value of x, the derivative of 9x remains 9.

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

• Linear Function: A function of the form f(x) = ax + b, where the graph is a straight line.

• Constant Rule: A rule stating that the derivative of a constant is zero.

• Coefficient Rule: A rule stating that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.

• Power Rule: A basic differentiation rule used to find the derivative of a power of x, such as xⁿ.