Derivative of 6^x

We now understand the derivative of 6^x. It is commonly represented as d/dx (\(6^x\)) or (\(6^x\))', and its value is \(6^x\) ln(6).

The function\( 6^x \) has a clearly defined derivative, indicating it is differentiable for all real numbers x.

The key concepts are mentioned below:

Exponential Function: (\(6^x\)).

Logarithmic Function: ln(x) is the natural logarithm of x.

Chain Rule: Rule for differentiating composite functions such as \(a^x\).

The derivative of \(6^x\) can be denoted as d/dx (\(6^x\)) or (\(6^x\))'. The formula we use to differentiate \(6^x\) is: d/dx (\(6^x\)) = \(6^x\) ln(6) The formula applies to all x.

We can derive the derivative of 6^x using proofs.

To show this, we will use the properties of exponential functions along with the rules of differentiation.

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

By First Principle

Using Chain Rule

Using Logarithmic Differentiation

We will now demonstrate that the differentiation of 6^x results in 6^x ln(6) using the above-mentioned methods:

By First Principle

The derivative of \(6^x\) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of\( 6^x\) using the first principle, we will consider f(x) =\( 6^x\).

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

Given that f(x) = \(6^x\), we write f(x + h) = \(6^(x + h)\).

Substituting these into equation (1), f'(x) = limₕ→₀ [\(6^(x + h) - 6^x\)] / h = limₕ→₀ [\(6^x (6^h - 1)\)] / h = \(6^x\) limₕ→₀ [\((6^h - 1) / h\)]

Using the limit property, limₕ→₀ [\((6^h - 1) / h\)] = ln(6). f'(x) = \(6^x\) ln(6)

Hence, proved.

Using Chain Rule

To prove the differentiation of 6^x using the chain rule, We express 6^x as e^(x ln(6)).

Let u = x ln(6). Then, 6^x = e^u. By the chain rule: d/dx [e^u] = e^u (du/dx)

So we get, d/dx (6^x) = e^(x ln(6)) * ln(6)

Substituting back the expression for e^(x ln(6)) gives us: d/dx (6^x) = 6^x ln(6).

Using Logarithmic Differentiation

We will now prove the derivative of \(6^x \)using logarithmic differentiation.

The step-by-step process is demonstrated below: Consider y = \(6^x\)

Taking the natural logarithm of both sides gives ln(y) = ln(\(6^x\)) = x ln(6)

Differentiating both sides with respect to x gives: (1/y) dy/dx = ln(6)

Thus, dy/dx = y ln(6) = \(6^x \)ln(6)

Therefore, the derivative of the function y =\( 6^x\) is \(6^x\) ln(6).

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 \(6^x\).

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 6^x, 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 derivative of \(6^x\) is defined for all real numbers x since it is an exponential function. At x = 0, the derivative of \(6^x = 6^0 \)ln(6), which is ln(6).

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

•  Natural Logarithm: The logarithm to the base e, denoted as ln(x), where e is approximately equal to 2.71828.

•  Chain Rule: A rule for differentiating composite functions, used to differentiate expressions like (f(g(x))).

•  Quotient Rule: A rule for differentiating the division of two functions, used to find the derivative of u/v.

•  Logarithmic Differentiation: A method for differentiating functions by taking the natural logarithm of both sides and then differentiating.