How Much Facts Do You Know About Advanced Derivatives?

Explanation
The derivative of y with respect to x is defined as the change in y divided by the change in x. This is because the derivative measures the rate at which y changes with respect to x, and is represented as dy/dx. Therefore, the correct answer is "As the change in y over x."

Explanation
A linear function is characterized by a constant rate of change. This means that the function always increases or decreases by the same amount for every unit increase in the independent variable. Unlike other functions that may fluctuate or have varying rates of change, a linear function maintains a consistent pattern, making it constant.

Explanation
The given answer, "ax+b," is a common form of a linear function. In this form, "a" represents the slope of the line, and "b" represents the y-intercept. The slope determines how steep the line is, and the y-intercept is the point where the line crosses the y-axis. By plugging in different values for "x," you can find corresponding values for "y" and plot them on a graph to create a linear function.

Explanation
A power function is a mathematical function in the form of y = ax^b, where a and b are constants. The particularity of a power function is that its slope varies. The slope of a power function depends on the value of the exponent b. If b is positive, the slope increases as x increases. If b is negative, the slope decreases as x increases. Therefore, the slope of a power function is not constant and can vary depending on the value of the exponent.

Explanation
The derivative of cosine is a negative sine. This can be derived using the chain rule and the derivative of sine.

Explanation
A cosine function is a mathematical function that represents the ratio of the adjacent side to the hypotenuse in a right triangle. It is not a variant or a function of the sine, nor is it 1/2 of the sine. However, the cosine function can be derived from the sine function by taking the derivative. This means that the rate of change of the sine function at any point is equal to the cosine function at that point.

Explanation
Derivatives are commonly used in Newton's method, which is an iterative numerical method for finding the roots of an equation. In this method, derivatives are used to approximate the slope of the function at a given point, allowing for the calculation of more accurate and efficient approximations of the roots. Therefore, derivatives find their use specifically in Newton's method.

Explanation
The given expression (d/dx) . (3x^6 + x^2 - 6) can be broken up into (d/dx) (3x^6) + (d/dx) (x^2) - (d/dx) (6). This is because the derivative operator (d/dx) can be distributed over each term in the expression, resulting in the derivatives of each individual term being taken separately. Therefore, the correct answer is (d/dx) (3x^6) + (d/dx) (x^2) - (d/dx) (6).

Explanation
Infinitesimal is the best term to describe the small distance between two points x0 and x1. It refers to a quantity that is extremely small, approaching zero. In mathematics, infinitesimals are used to represent values that are too small to be measured or calculated precisely. Therefore, infinitesimal accurately captures the concept of a very small distance between two points, emphasizing the idea of it being almost negligible or approaching zero.

Explanation
The rule for power functions when taking the derivative is d/dx (x^a)= ax ^(a-1). This means that when you differentiate a power function, you bring down the exponent as a coefficient and subtract 1 from the exponent.