Derivative of Root 3x

The derivative of root 3x is commonly represented as d/dx [(3x)^(1/2)] or [(3x)^(1/2)]', and its value is (3/2)(3x)^(-1/2). The function (3x)^(1/2) has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Square Root Function: ((3x)^(1/2) indicates the square root of 3x). Power Rule: Rule for differentiating (3x)^(1/2).

The derivative of root 3x can be denoted as d/dx [(3x)^(1/2)] or [(3x)^(1/2)]'. The formula used to differentiate root 3x is: d/dx [(3x)^(1/2)] = (3/2)(3x)^(-1/2) The formula applies to all x where x > 0.

We can derive the derivative of root 3x using proofs. To show this, we will use the rules of differentiation. There are several methods we use to prove this, such as: By First Principle Using Chain Rule Using the Power Rule We will now demonstrate that the differentiation of (3x)^(1/2) results in (3/2)(3x)^(-1/2) using the above-mentioned methods: By First Principle The derivative of (3x)^(1/2) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of (3x)^(1/2) using the first principle, we will consider f(x) = (3x)^(1/2). Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = (3x)^(1/2), we write f(x + h) = (3(x + h))^(1/2). Substituting these into equation (1), f'(x) = limₕ→₀ [(3(x + h))^(1/2) - (3x)^(1/2)] / h Using binomial expansion and simplification, f'(x) = (3/2)(3x)^(-1/2). Hence, proved. Using Chain Rule To prove the differentiation of (3x)^(1/2) using the chain rule, Let u = 3x, then f(x) = u^(1/2). By the chain rule, d/dx [u^(1/2)] = (1/2)u^(-1/2) * (du/dx). Since du/dx = 3, d/dx [(3x)^(1/2)] = (1/2)(3x)^(-1/2) * 3 = (3/2)(3x)^(-1/2). Using the Power Rule We will now prove the derivative of (3x)^(1/2) using the power rule. f(x) = (3x)^(1/2) can be expressed as (3x)^0.5. Using the power rule, d/dx [u^n] = n*u^(n-1) * (du/dx). We have u = 3x, n = 0.5, and du/dx = 3. d/dx [(3x)^(1/2)] = 0.5*(3x)^(-0.5) * 3 = (3/2)(3x)^(-1/2).

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 (3x)^(1/2). 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 (3x)^(1/2), 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 x = 0, the derivative is undefined because (3x)^(1/2) is not defined for x = 0. When x = 1, the derivative of (3x)^(1/2) is (3/2)(3)^(1/2).

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Square Root Function: A function involving the square root of a variable, such as (3x)^(1/2). Power Rule: A basic rule for differentiation used for functions in the form of xⁿ, where n is a real number. Chain Rule: A differentiation rule used to differentiate the composition of functions. Higher-Order Derivative: The result of differentiating a function multiple times, giving insights into its changing rates.