Differentiation Formulas

General Formulas

Constant Rule: For any constant c, dc = 0.
If a quantity does not change, then its differential is zero.

Constant Multiple Rule: For any constant c and any quantity u, d(cu)=cdu.
The differential of a constant times a quantity is the constant times the differential of the quantity.

Sum Rule: d(u + v) = du + dv.
The differential of the sum of two quantities is the sum of their differentials.

Product Rule: d(uv) = udv + vdu.
The differential of the product of two quantities is the first quantity times the differential of the second quantity plus the second quantity times the differential of the first quantity.

Quotient Rule: d(u/v) = (vdu - udv)/v^2.
The differential of a quotient of two quantities is the denominator quantity times the differential of the numerator quantity minus the numerator quantity times the differential of denominator quantity, all divided by the square of the denominator quantity.

Power Rule: For any (rational) constant n and any quantity u, d(u^n) = nu^(n - 1)du.
The differential of a quantity raised to a power is the power times the quantity raised to the original power minus 1, times the differential of the quantity.

Trigonometric Formulas

d(sin(u)) = cos(u)du
The differential of the sine of a quantity is the cosine of the quantity times the differential of the quantity.

d(cos(u)) = -sin(u)du
The differential of the cosine of a quantity is the negative of the sine of the quantity times the differential of the quantity.

d(tan(u)) = (sec(u))^2du
The differential of the tangent of a quantity is the square of the secant of the quantity times the differential of the quantity.

d(sec(u)) = sec(u)tan(u)du
The differential of the secant of a quantity is the secant of the quantity times the tangent of the quantity times the differential of the quantity.

Inverse Trigonometric Formulas

d(arctan(u)) = du/(1 + u^2)
The differential of the inverse tangent of a quantity is the reciprocal of 1 plus the square of the quantity, times the differential of the quantity.

d(arcsin(u)) = du/sqrt(1 - u^2)
The differential of the inverse sine of a quantity is the reciprocal of the square root of 1 minus the square of the quantity, times the differential of the quantity.

d(arcsec(u)) = du/(u*sqrt(u^2 - 1))
The differential of the inverse secant of a quantity is the reciprocal of the product of the quantity and the square root of the square of the quantity minus 1, times the differential of the quantity.

Exponential Function Formulas

d(exp(u)) = exp(u)du or, letting e = exp(1), d(e^u) = (e^u)du
The differential of e raised to a quantity is e raised to that quantity times the differential of the quantity.

For any positive base b, (b not equal to 1), d(b^u) = b^u(ln(b))du
The differential of b raised to a quantity is b raised to that quantity, times the natural logarithm of b, times the differential of the quantity.

Logarithm Function Formulas

d(ln(u)) = du/u
The differential of the natural logarithm of a quantity is the reciprocal of the quantity times the differential of the quantity.

For any positive base b, (b not equal to 1), d(log_b(u)) = du/(uln(b))
The differential of the base-b logarithm of a quantity is the reciprocal of the quantity times the natural logarithm of b, times the differential of the quantity.