Derivatives

Derivative problems on the test almost always involve the power rule for polynomials. For a term $ax^n$, its derivative is $anx^{n-1}$. To find the derivative of a polynomial, apply this rule to each term.

For $f(x) = x^3 - 3x^2 + x - 3$, the first derivative is $f'(x) = 3x^2 - 6x + 1$. To find the second derivative, $f''(x)$, simply differentiate $f'(x)$: $f''(x) = 6x - 6$.

Occasionally, derivatives of other functions like sin, cos, e^x, and ln x are needed, especially for L'Hôpital's rule.