Quiz On Derivatives Introduction

Derivatives are mathematical tools used to find the rate of change of a function at a specific point. By calculating the derivatives, we can determine the maximum and minimum points of a function, also known as maximas and minimas. This is the most obvious and common use of derivatives in various fields such as economics, physics, and engineering. It allows us to optimize processes, find critical points, and make informed decisions based on the behavior of a function.

The derivative of the logarithm of x is 1/x. This can be derived using the power rule of differentiation. The power rule states that the derivative of x^n is equal to n*x^(n-1). In this case, the base of the logarithm is e (the natural logarithm) and the exponent is 1. Therefore, the derivative of ln(x) is 1/x.

The given expression can be broken up by factoring out the common factors from the polynomial. In this case, the common factor is x. By factoring out x, we get x(3x^5 + 1 - 6/x). Simplifying further, we get 3x^6 + x^2 - 6. Therefore, the correct answer is 18x^5 + 2x.

The symbol "∬" represents a double integral. In mathematics, a double integral is used to calculate the volume or area of a two-dimensional region in space. It involves integrating a function over a region in the xy-plane, where the region is defined by two sets of boundaries. The symbol "∬" is a shorthand notation for denoting this type of integral.

The correct answer is f'(x) = ae^ax. This is the correct answer because when we differentiate the function f(x) = e^ax with respect to x, we use the chain rule and the derivative of e^ax is ae^ax. Therefore, the derivative of f(x) is ae^ax.

The derivative of tan x is equal to 1 divided by the square of the cosine of x. This can be derived using the quotient rule for differentiation.

The formula for the equation of the tangent is y=f(x0) + f'(x0). (x-x0). This formula combines the function value at a specific point (x0) with the derivative of the function at that point (f'(x0)), and the difference between the x-coordinate of the point of tangency (x) and the x-coordinate of the specific point (x0). This formula allows us to find the equation of the tangent line to a curve at a specific point.

The correct answer is d
__ (∫f(x) dx) = f(x)
dx

This is the correct formula for the derivative of an integral. It states that the derivative of an integral of a function f(x) with respect to x is equal to the original function f(x). This is a fundamental property of integrals and derivatives in calculus.

The equation for arsin(x) is sin (arcsin x) = x. This equation represents the inverse relationship between the sine function and its inverse, arcsine. It states that if you take the arcsine of a value x and then take the sine of that result, you will obtain the original value x. This equation is derived from the definition of the arcsine function as the inverse of the sine function.

The derivative of a function can have two intuitions. The first intuition is that the derivative represents the slope of the function at a given point, indicating how fast the function is changing at that point. The second intuition is that the derivative can also represent the rate of change of the function over a certain interval, indicating the average rate at which the function is changing over that interval. These two intuitions help us understand the behavior and properties of functions in calculus.