Derivative of 10x

We now understand the derivative of 10x. It is commonly represented as d/dx (10x) or (10x)', and its value is 10. The function 10x has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: - Linear Function: A function of the form f(x) = mx + b. - Constant Rule: The derivative of a constant multiplied by a function. - Power Rule: Used for differentiating polynomials.

The derivative of 10x can be denoted as d/dx (10x) or (10x)'. The formula we use to differentiate 10x is: d/dx (10x) = 10 The formula applies to all x since it is a linear function with no restrictions on its domain.

We can derive the derivative of 10x using proofs. To show this, we will use the basic rules of differentiation. There are several methods we use to prove this, such as: - By First Principle - Using the Constant Rule - Using the Power Rule We will now demonstrate that the differentiation of 10x results in 10 using the above-mentioned methods: By First Principle The derivative of 10x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of 10x using the first principle, we will consider f(x) = 10x. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h Given that f(x) = 10x, we write f(x + h) = 10(x + h). Substituting these into the equation, f'(x) = limₕ→₀ [10(x + h) - 10x] / h = limₕ→₀ [10x + 10h - 10x] / h = limₕ→₀ 10h / h = limₕ→₀ 10 f'(x) = 10. Hence, proved. Using the Constant Rule To prove the differentiation of 10x using the constant rule, We use the formula: d/dx (c·f(x)) = c·d/dx (f(x)) where c is a constant and f(x) = x. Applying this rule, d/dx (10x) = 10·d/dx (x) Since d/dx (x) = 1, d/dx (10x) = 10·1 = 10. Hence, proved. Using the Power Rule We will now prove the derivative of 10x using the power rule. The step-by-step process is demonstrated below: The power rule formula is d/dx (x^n) = n·x^(n-1). Given f(x) = 10x, we can rewrite it as 10x^1. Applying the power rule, d/dx (10x) = 10·1·x^(1-1) = 10·1·x^0 = 10·1·1 = 10. Hence, proved.

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be straightforward for linear functions. To understand them better, think of a car where the speed changes (first derivative), and the rate at which the speed changes (second derivative) remains constant. Higher-order derivatives help us understand functions like 10x more deeply. 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, f′′(x), is derived from the first derivative, but for linear functions like 10x, it is zero, indicating no change in the rate of change. Similarly, the third derivative, f′′′(x), and beyond are also zero.

Since 10x is a linear function, there are no special cases where the derivative is undefined. The derivative is always 10 for all x.

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Linear Function: A function with a constant rate of change, typically in the form f(x) = mx + b. Constant Rule: A rule stating that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. Power Rule: A rule used to find the derivative of a function of the form x^n, resulting in n·x^(n-1). First Derivative: The initial result of a function, which gives us the rate of change of a specific function.