Derivative of a^x

We now understand the derivative of a^x. It is commonly represented as d/dx (a^x) or (a^x)', and its value is a^x ln(a). The function a^x has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Exponential Function: (a^x). Chain Rule: Rule for differentiating composite functions like a^x. Natural Logarithm: ln(a) is the natural logarithm of a.

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

We can derive the derivative of a^x using proofs. To show this, we will use the definition of the derivative and logarithmic differentiation. There are several methods we use to prove this, such as: By First Principle Using Logarithmic Differentiation We will now demonstrate that the differentiation of a^x results in a^x ln(a) using the above-mentioned methods: By First Principle The derivative of a^x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of a^x using the first principle, we will consider f(x) = a^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) = a^x, we write f(x + h) = a^(x + h). Substituting these into equation (1), f'(x) = limₕ→₀ [a^(x + h) - a^x] / h = limₕ→₀ [a^x (a^h - 1)] / h = a^x limₕ→₀ [(a^h - 1) / h] Using the property of exponential functions and limits, limₕ→₀ [(a^h - 1) / h] = ln(a). Thus, f'(x) = a^x ln(a). Hence, proved. Using Logarithmic Differentiation To prove the differentiation of a^x using logarithmic differentiation, Consider y = a^x. Taking the natural logarithm of both sides, ln(y) = x ln(a). Differentiating both sides with respect to x, d/dx [ln(y)] = d/dx [x ln(a)]. Using the derivative of ln(y), we have 1/y dy/dx = ln(a). Thus, dy/dx = y ln(a). Substituting y = a^x, we have, dy/dx = a^x ln(a). 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 a^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 a^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).

When the base a is 1, the derivative is zero because any power of 1 is still 1, which is constant. When the base a is e (Euler's number), the derivative of e^x = e^x ln(e), which simplifies to e^x since ln(e) = 1.

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Exponential Function: An exponential function is a mathematical function in the form a^x, where a is a constant. Natural Logarithm: The natural logarithm, denoted as ln, is the logarithm to the base e (Euler's number). Chain Rule: A rule in calculus for differentiating the composition of two or more functions. Quotient Rule: A method for differentiating the division of two functions.