Derivative of \(5^{2x}\)

We now understand the derivative of (5^{2x}). It is commonly represented as (frac{d}{dx} (5^{2x})) or ((5^{2x})'), and its value is (2 × 5^{2x} ln(5)). The function (5^{2x}) has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Exponential Function: (5^{2x}) is an exponential function with base 5.

Chain Rule: Rule for differentiating (5^{2x}) because it involves a function of a function.

Logarithmic Function: The natural logarithm, (ln), is used when differentiating exponential functions.

The derivative of (5^{2x}) can be denoted as (frac{d}{dx} (5^{2x})) or ((5^{2x})').

The formula we use to differentiate (5^{2x}) is: (frac{d}{dx} (5^{2x}) = 2 times 5^{2x} ln(5))

This formula applies to all (x) as the exponential function is defined for all real numbers.

We can derive the derivative of (5^{2x}) using proofs. To show this, we will use the rules of differentiation along with properties of exponential functions.

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

• Using Chain Rule
   
• Using Logarithmic Differentiation

We will now demonstrate that the differentiation of (5^{2x}) results in (2 times 5^{2x} ln(5)) using the above-mentioned methods:

Using Chain Rule

To prove the differentiation of (5^{2x}) using the chain rule, Consider (f(x) = 5^{2x}). Let (u = 2x), then (f(x) = z^u). The derivative of (5^u) with respect to (u) is (5^u ln(5)). Now, (frac{du}{dx} = 2). By the chain rule, (frac{d}{dx} (5^{2x}) = frac{d}{du} (5^u) cdot frac{du}{dx}). (frac{d}{dx} (5^{2x}) = 5^{2x} ln(5) cdot 2 = 2 times 5^{2x} ln(5)). Hence, proved.

Using Logarithmic Differentiation

To prove the differentiation of (5^{2x}) using logarithmic differentiation, Let (y = 5^{2x}). Take the natural logarithm of both sides, (ln(y) = ln(5^{2x})). Using properties of logarithms, (ln(y) = 2x ln(5)). Differentiate implicitly: (frac{1}{y} frac{dy}{dx} = 2 ln(5)). Therefore, (frac{dy}{dx} = y cdot 2 ln(5)). Substitute back (y = 5^{2x}), (frac{dy}{dx} = 5^{2x} cdot 2 ln(5)). Thus, (frac{d}{dx} (5^{2x}) = 2 times 5^{2x} ln(5)).

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 (5^{2x}).

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 (5^{2x}), 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 (continuing for higher-order derivatives). The nth derivative of (5^{2x}) will be ((2 ln(5))ⁿ times 5^{2x}).

When (x = 0), the derivative of (5^{2x}) is (2 ln(5)), which simplifies to (2 ln(5)) since (5^{0} = 1).

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

• Exponential Function: A function of the form (a^{x}), where (a) is a constant base and (x) is the exponent.

• Chain Rule: A rule for differentiating compositions of functions, used in differentiating exponential functions.

• Logarithmic Function: A function related to the inverse of the exponential function, often used in differentiation of exponentials.

• Quotient Rule: A rule for differentiating ratios of functions, used when differentiating (frac{5^{2x}}{x}).