Summary
Highlights
The derivative of any constant is always zero. The power rule states that the derivative of x^n is n*x^(n-1). For functions where a constant is raised to a variable (a^x), the derivative is a^x * Ln(a). If the exponent is a function (a^u), the derivative becomes a^u * U' * Ln(a). For variables raised to variables, logarithmic differentiation is required.
The constant multiple rule states that the derivative of c*f(x) is c times the derivative of f(x). The product rule for two functions u and v is (u'v + uv'). The quotient rule for u/v is (vu' - uv') / v^2.
The chain rule is used for composite functions. For f(G(U)), the derivative is f'(G(U)) * G'(U) * U'. If dealing with f(G(x)), the derivative is f'(G(x)) * G'(x).
When combining the chain rule with the power rule, if you have f(x)^n, the derivative is n*f(x)^(n-1) * f'(x). Another form of the chain rule is dy/dx = (dy/du) * (du/dx).
For log base a of U, the derivative is U' / (U * Ln(a)). For the natural logarithm of U (Ln U), the derivative is U' / U, as Ln(e) equals one.
The derivatives are: sin(U) = cos(U) * U'; cos(U) = -sin(U) * U'; tan(U) = sec^2(U) * U'; cot(U) = -csc^2(U) * U'; sec(U) = sec(U)tan(U) * U'; csc(U) = -csc(U)cot(U) * U'.
Important inverse trig derivatives include: arcsin(U) = U' / sqrt(1 - U^2); arccos(U) = -U' / sqrt(1 - U^2); arctan(U) = U' / (1 + U^2); arccot(U) = -U' / (1 + U^2); arcsec(U) = U' / (U * sqrt(U^2 - 1)); arccsc(U) = -U' / (U * sqrt(U^2 - 1)).