General Knowledge and Calculus Basics

The statement '4 is even AND 8 is odd' uses the AND logical connective. For 'A AND B' to be true, BOTH parts must be true. Here, '4 is even' is true, but '8 is odd' is false (8 is even). Since one part is false, the entire AND statement is false. This is a fundamental property of logical conjunction.

The integral of sin(x) dx is -cos(x) + C. This is because the derivative of cos(x) is -sin(x), so the derivative of -cos(x) is sin(x). Positive cos(x) is the derivative of sin(x), not its integral.

The integral of cos(x) dx is sin(x) + C. Since the derivative of sin(x) is cos(x), the antiderivative (integral) of cos(x) must be sin(x). A common mistake is to suggest -sin(x), which is actually the derivative of cos(x).

The derivative of cos(x) is -sin(x). This follows from the rules of trigonometric differentiation. Conversely, the derivative of sin(x) is positive cos(x). Distractors like -cos(x) or -tan(x) do not follow the standard differentiation rules for this function.

In calculus, the derivative (differential) of sin(x) is cos(x). This is a fundamental trigonometric identity. The derivative of cos(x) is -sin(x), which is a common point of confusion for students.

The derivative of the exponential function e^x with respect to x is e^x. This is a unique property where the rate of change of the function is equal to the value of the function itself. Sin x and Cos x are derivatives of trigonometric functions, not exponentials.