Derivative of 3 Square Root of 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 (3/2)x^(-1/2). The function 3√x has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Square Root Function: (√x = x^(1/2)). Power Rule: Rule for differentiating x^(n) (since it involves powers of x). Constant Multiple Rule: When a constant multiplies a function, differentiate the function and multiply by the constant.

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) = (3/2)x^(-1/2) The formula applies to all x > 0.

We can derive the derivative of 3√x using proofs. To show this, we will use the power rule along with the rules of differentiation. There are several methods we use to prove this, such as: By First Principle Using Power Rule Using Constant Multiple Rule We will now demonstrate that the differentiation of 3√x results in (3/2)x^(-1/2) 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 = 3 limₕ→₀ [√(x + h) - √x] / h Multiply by the conjugate to simplify: = 3 limₕ→₀ [(x + h - x) / h(√(x + h) + √x)] = 3 limₕ→₀ h / [h(√(x + h) + √x)] = 3 limₕ→₀ 1 / (√(x + h) + √x) = 3/(2√x) As x approaches 0, the expression simplifies to (3/2)x^(-1/2). Using Power Rule To prove the differentiation of 3√x using the power rule, We use the formula: 3√x = 3x^(1/2) Using the power rule: d/dx [x^n] = n*x^(n-1) d/dx (3x^(1/2)) = 3 * (1/2) * x^(-1/2) = (3/2) * x^(-1/2) Using Constant Multiple Rule We will now prove the derivative of 3√x using the constant multiple rule. The step-by-step process is demonstrated below: Here, we use the formula, 3√x = 3 * x^(1/2) Using the constant multiple rule where d/dx [c*f(x)] = c*f'(x), d/dx (3√x) = 3 * d/dx (x^(1/2)) Using the power rule: d/dx (x^(1/2)) = (1/2) * x^(-1/2) So, d/dx (3√x) = 3 * (1/2) * x^(-1/2) = (3/2) * x^(-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 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 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 is 0, the derivative is undefined because the square root function is not defined for non-positive values of x. When x is 1, the derivative of 3√x = (3/2)x^(-1/2), which is 3/2.

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 that involves the root of a number, typically represented as √x. Power Rule: A rule used to differentiate functions of the form x^n, where n is a constant. First Derivative: The initial result of a function, which gives us the rate of change of a specific function. Constant Multiple Rule: A rule stating that when a constant multiplies a function, the derivative is the constant multiplied by the derivative of the function.