As of this year, the derivative of 1/x is still -1/x². This can be seen in various online sources such as Math is Fun, Khan Academy, and Calculus How To

How

The derivative of 1/x can be derived using the power rule of differentiation. Since 1/x can be expressed as x^-1, we can apply the power rule to get:

• • d/dx (x^-1) = -1x^-1-1 = -1/x²

Thus, the derivative of 1/x is -1/x².

For instance, consider the function y = 1/x. To find the slope of the tangent line at a particular point, say x = 2, we can find the derivative as follows:

• • y’ = d/dx (1/x) = -1/x²

• • y'(2) = -1/2² = -1/4

Hence, the slope of the tangent line to the graph of y = 1/x at x = 2 is -1/4.

Why

The reason the derivative of 1/x is -1/x² lies in the fact that the slope of the tangent line to the graph of y = 1/x at any point (a,b) is given by the limit:

• • lim (x → a) [(1/x) – (1/a)] / (x – a)

If we apply L’Hopital’s rule to this limit, we end up with -1/a², which is the same as -1/x² since a = x in this case. Thus, the derivative of 1/x is -1/x² for all x.

Or

Currently, the derivative of 1/x remains the same as before. However, it is worth noting that the concept of derivatives is constantly evolving in mathematics and related fields, and new research may uncover further insights into this topic.

Who

The derivative of 1/x is a fundamental concept in calculus and is relevant to students and professionals in various fields such as mathematics, physics, and engineering. It is also important for anyone who wants to understand the behavior of functions and their graphs.

Additional Questions:

1. 1. What is the derivative of ln(x)

1. 1. • As of this year, the derivative of ln(x) is 1/x. This can be found in various sources such as Math is Fun, Khan Academy, and Calculus How To.

1. 2. What is the derivative of e^x

1. 1. • The derivative of e^x is e^x. This can be found in various sources such as Math is Fun, Khan Academy, and Calculus How To.

1. 2. What is the derivative of sin(x)

1. 1. • The derivative of sin(x) is cos(x). This can be found in various sources such as Math is Fun, Khan Academy, and Calculus How To.

1. 2. What is the second derivative of f(x) = x³

1. 1. • The first derivative of f(x) = x³ is f'(x) = 3x². The second derivative of f(x) is the derivative of f'(x), which is f »(x) = d/dx [3x²] = 6x. Thus, the second derivative of f(x) = x³ is 6x.

1. 2. What is the derivative of ln(2x)

1. 1. • The derivative of ln(2x) is 1/x. This can be found in various sources such as Math is Fun, Khan Academy, and Calculus How To.

1. 2. What is the derivative of cos(x)

1. 1. • The derivative of cos(x) is -sin(x). This can be found in various sources such as Math is Fun, Khan Academy, and Calculus How To.

1. 2. What is the derivative of x² + 3x – 4

1. 1. • The derivative of x² is 2x. The derivative of 3x is 3. The derivative of a constant is 0. Therefore, the derivative of f(x) = x² + 3x – 4 is f'(x) = 2x + 3.

1. 2. What is the first derivative test

1. 1. • The first derivative test is a method for finding relative extrema of a function. It involves finding the critical points of the function (where the derivative is equal to zero or undefined) and then testing the sign of the derivative in intervals between these points. If the sign of the derivative changes from positive to negative, then the function has a relative maximum at that point. If the sign changes from negative to positive, then the function has a relative minimum at that point. If the sign does not change, then the point is an inflection point. This can be found in various sources such as Math is Fun, Khan Academy, and Calculus How To.

1. 2. What is the second derivative test

1. 1. • The second derivative test is a method for determining the nature of extrema (whether they are maximum or minimum) of a function. It involves finding the critical points of the function (where the first derivative is equal to zero or undefined) and then evaluating the second derivative at these points. If the second derivative is positive at a critical point, then the function has a relative minimum at that point. If the second derivative is negative, then the function has a relative maximum at that point. If the second derivative is zero, then the test is inconclusive. This can be found in various sources such as Math is Fun, Khan Academy, and Calculus How To.

Sources:

• • Math is Fun. (n.d.). Derivatives: Introduction. Retrieved June 19, 2023.

• • Khan Academy. (n.d.). Differential Calculus. Retrieved June 19, 2023.

• • Calculus How To. (n.d.). Derivative Formulas. Retrieved June 19, 2023.