Calculus Formulas Study Guide

1. Derivative Formulas

Constant Rule:

If f(x) = c, where c is a constant, then the derivative f'(x) = 0.

Power Rule:

If f(x) = x^n, where n is a constant, then the derivative f'(x) = nx^(n-1).

Product Rule:

If f(x) = u(x)v(x), then the derivative f'(x) = u'v + uv'.

Quotient Rule:

If f(x) = u(x)/v(x), then the derivative f'(x) = (u'v - uv') / v^2.

2. Integral Formulas

Power Rule for Integration:

The integral of x^n dx = (1/(n+1))x^(n+1) + C, where C is the constant of integration.

Integration by Parts:

The formula for integration by parts is ∫u dv = uv - ∫v du, where u and v are differentiable functions of x.

Integration by Substitution:

If ∫f(g(x))g'(x) dx, then let u = g(x) and du = g'(x) dx. This helps in simplifying the integral.

3. Limit Formulas

Limit of a Constant:

The limit of a constant c as x approaches a is lim(x→a) c = c.

Limit of a Polynomial:

The limit of a polynomial function f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0 as x approaches a is lim(x→a) f(x) = f(a).

Limit of a Rational Function:

The limit of a rational function f(x) = g(x)/h(x) as x approaches a is lim(x→a) f(x) = g(a)/h(a), provided that h(a) ≠ 0.