Question

Write two different explanations of the relationship between the expressions 4x3 and x4, one using the word derivative and the other using the word antiderivative. Then give an example of a pair of expressions that have the same relationship.

Answer 1

Given expressions $4x^3 and x^4$

Derivative

For derivative we multiply the exponent and then subtract 1 from the exponent

Derivative of $x^4$ is 4 * $x^3$ = $4x^3$

For anti derivative we add 1 to the exponent and then divide the same exponent

Anti-derivative of $4x^3$ is 4 * $\frac{x^4}{4}$ = $x^4$

Antiderivative of $4x^3$ is $x^4$

Derivative of $x^4$ is $4x^3$

Example expressions

2x and $x^2$

Antiderivative of $2x$ is $x^2$

Derivative of $x^2$ is $2x$