The derivative of arccot x is closely related to the derivative of arctan x, the derivative of cot x, the derivative of sec^2 x, and the derivative of csc^2 x. These derivatives share similar formulas and have applications in calculus and trigonometry. Understanding the derivative of arccot x involves examining its relationship to these other derivatives and their respective functions.
How Do You Differentiate arccot x?
The derivative of arccot x is -1/(1+x^2).
In general, the derivative of arccot x is found using the formula:
d/dx arccot x = -1/(1+x^2)
This formula can be derived using the chain rule and the derivative of cot x:
d/dx arccot x = 1/(1+cot^2 x) * d/dx cot x
= 1/(1+cot^2 x) * (-csc^2 x)
= -1/(1+cot^2 x) * (1/sin^2 x)
= -1/(1+cot^2 x) * (1/(1+cos^2 x))
= -1/(1+x^2)
Here is a table summarizing the derivative of arccot x:
| Function | Derivative |
|---|---|
| arccot x | -1/(1+x^2) |
Additionally, here are some examples of how to differentiate arccot x:
d/dx arccot (2x) = -1/(1+(2x)^2) = -1/(4x^2+1)
d/dx arccot (x^3) = -1/(1+(x^3)^2) = -1/(x^6+1)
d/dx arccot (sin x) = -1/(1+(sin x)^2) = -1/(1+sin^2 x)
Question 1: How to determine the derivative of the inverse cotangent function?
Answer: The derivative of the arccotangent function, denoted as arccot(x), can be calculated using the formula: d/dx arccot(x) = -1 / (1 + x^2). This formula is derived from the chain rule and the derivative of the cotangent function.
Question 2: What is the relationship between the derivative of arccot(x) and the derivative of cot(x)?
Answer: The derivative of arccot(x) is inversely proportional to the derivative of cot(x). Specifically, d/dx arccot(x) = -1 / (d/dx cot(x)). This relationship highlights the inverse function relationship between arccot(x) and cot(x).
Question 3: What are the key steps involved in calculating the derivative of arccot(x) using the chain rule?
Answer: To calculate the derivative of arccot(x) using the chain rule, the following steps are involved:
– Identify the inner and outer functions: The outer function is arccot(x), and the inner function is x.
– Find the derivative of the inner function: d/dx x = 1.
– Find the derivative of the outer function: d/dx arccot(x) = -1 / (1 + x^2).
– Combine the results: Applying the chain rule, we get d/dx arccot(x) = -1 * (1 / (1 + x^2)) * (1), which simplifies to -1 / (1 + x^2).
Well, that’s a wrap on the enigmatic arccot x and its slippery slope of a derivative. I hope you found this mathematical escapade both illuminating and slightly mind-boggling. To those who ventured into this article out of pure curiosity, I commend your bravery. And to those who came seeking elucidation, I hope I’ve shed some light on this geometrical paradox. As you bid farewell, remember the magical formula (-1/1+x^2) and the newfound respect you’ve gained for the complexities of math. Drop by again soon for more mathemagical adventures where we’ll tackle even quirkier functions and unravel their hidden secrets. Until then, stay curious, my friends!