Derivative of 10/x

We now understand the derivative of 10/x. It is commonly represented as d/dx (10/x) or (10/x)', and its value is -10/x².

The function 10/x has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below:

Reciprocal Function: (10/x).

Power Rule: Rule for differentiating x raised to a power.

Negative Exponents: Understanding how to differentiate functions with negative exponents.

The derivative of 10/x can be denoted as d/dx (10/x) or (10/x)'.

The formula we use to differentiate 10/x is: d/dx (10/x) = -10/x² (or) (10/x)' = -10/x²

The formula applies to all x where x ≠ 0.

We can derive the derivative of 10/x using proofs. To show this, we will use basic differentiation rules. There are several methods we use to prove this, such as:

1. By First Principle
2. Using Power Rule
3. Using Quotient Rule

We will now demonstrate that the differentiation of 10/x results in -10/x² using the above-mentioned methods:

By First Principle

The derivative of 10/x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of 10/x using the first principle, we will consider f(x) = 10/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) = 10/x, we write f(x + h) = 10/(x + h).

Substituting these into equation (1),

f'(x) = limₕ→₀ [10/(x + h) - 10/x] / h = limₕ→₀ [10x - 10(x + h)] / [hx(x + h)] = limₕ→₀ [-10h] / [hx(x + h)] = limₕ→₀ [-10] / [x(x + h)] = -10/x²

Hence, proved.

Using Power Rule

To prove the differentiation of 10/x using the power rule, Rewrite 10/x as 10x⁻¹.

Using the power rule: d/dx (xⁿ) = nxⁿ⁻¹ d/dx (10x⁻¹) = 10(-1)x⁻² = -10x⁻²

This simplifies to -10/x². Using Quotient Rule We will now prove the derivative of 10/x using the quotient rule. The step-by-step process is demonstrated below: Let u = 10 and v = x.

By quotient rule: d/dx [u/v] = [v u' - u v'] / [v²]

Let’s substitute u = 10 and v = x, d/dx (10/x) = [x(0) - 10(1)] / x² = -10/x² Thus, d/dx (10/x) = -10/x².

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 10/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 10/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 10/x has a discontinuity there. When x is 1, the derivative of 10/x = -10/1², which is -10.

• Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.

• Reciprocal Function: A function in the form of a constant divided by a variable, such as 10/x.

• Power Rule: A basic rule in differentiation used to find the derivative of x raised to any power.

• Quotient Rule: A rule in differentiation used to find the derivative of the quotient of two functions.

• Undefined: Refers to the points where a function does not have a value, often leading to discontinuities.