Calculus

The antiderivative of 2x is found using the power rule, which yields x^2. We must include the arbitrary constant of integration C because the integral is indefinite, making option B correct. Option A is incorrect because it lacks this constant, representing only a single antiderivative rather than the general family.

This is a standard trigonometric limit where the ratio of the sine of an angle to the angle itself approaches 1 as the angle approaches 0. Option A is incorrect because, although the numerator approaches 0, the denominator also approaches 0, creating an indeterminate form that resolves to 1.

Applying the power rule of differentiation, we can directly find the derivative of x cubed. A diagram for this topic isn't available yet — the worked example below walks the same reasoning step by step.

• !Many students confuse the power rule of differentiation with the power rule of integration, which would result in x to the fourth, or they may forget to apply the exponent reduction by 1, leading to incorrect answers like x squared or 3x.
• !Option D is incorrect because x to the fourth is related to the integration of the function, not its differentiation.
• !Option A is incorrect because it lacks this constant, representing only a single antiderivative rather than the general family.
• !Option A is incorrect because, although the numerator approaches 0, the denominator also approaches 0, creating an indeterminate form that resolves to 1.
• !Option B is incorrect because a negative sign is only introduced when differentiating the cosine function, whereas the derivative of sine remains positive.

Answer each from memory — you'll see instantly whether you're right and why.

Run a focused 10-question mini-mock on Calculus and see it stick.