Derivative of 3^x

We now understand the derivative of 3^x. It is commonly represented as d/dx (3^x) or (3^x)', and its value is 3^x ln(3). The function 3^x has a clearly defined derivative, indicating it is differentiable for all real numbers. The key concepts are mentioned below: Exponential Function: (a^x, where a > 0). Exponential Rule: Rule for differentiating a^x (using a constant base). Natural Logarithm: ln(a) is the natural logarithm of a.

The derivative of 3^x can be denoted as d/dx (3^x) or (3^x)'. The formula we use to differentiate 3^x is: d/dx (3^x) = 3^x ln(3) (or) (3^x)' = 3^x ln(3) The formula applies to all x where 3^x is defined.

We can derive the derivative of 3^x using proofs. To show this, we will use the exponential and logarithmic identities 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 3^x results in 3^x ln(3) using the above-mentioned methods: By First Principle The derivative of 3^x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of 3^x using the first principle, we will consider f(x) = 3^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) = 3^x, we write f(x + h) = 3^(x + h). Substituting these into equation (1), f'(x) = limₕ→₀ [3^(x + h) - 3^x] / h = limₕ→₀ [3^x (3^h - 1)] / h = 3^x · limₕ→₀ [(3^h - 1) / h] Using the known limit limₕ→₀ [(3^h - 1) / h] = ln(3), we get: f'(x) = 3^x ln(3) Hence, proved. Using Chain Rule To prove the differentiation of 3^x using the chain rule, We use the formula: 3^x = e^(x ln(3)) Applying the chain rule: d/dx [e^(x ln(3))] = e^(x ln(3)) · ln(3) Since e^(x ln(3)) = 3^x, d/dx (3^x) = 3^x ln(3) Using Logarithmic Differentiation We will now prove the derivative of 3^x using logarithmic differentiation. The step-by-step process is demonstrated below: Consider y = 3^x Taking natural logarithms on both sides, ln(y) = x ln(3) Differentiating both sides with respect to x, d/dx [ln(y)] = d/dx [x ln(3)] (1/y) dy/dx = ln(3) dy/dx = y ln(3) Substituting y = 3^x, dy/dx = 3^x ln(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 3^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 3^x, 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).

3^x is differentiable for all real x, so there are no points where the derivative is undefined. The derivative of 3^x at x = 0 is simply 3^x ln(3) evaluated at x = 0, which is ln(3).

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 and x is a variable. Natural Logarithm: The logarithm to the base e, denoted as ln. Chain Rule: A rule for differentiating compositions of functions. Product Rule: A rule used to differentiate products of two functions.