Trigonometric derivatives
With x measured in radians, the derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). For tangent, d/dx tan(x) = 1/cos(x)^2, which is also written sec^2(x). These formulas are only this tidy in radians, which is why calculus always uses them.
Exponential functions
The number e is about 2.718 and is special because the function e^x is its own derivative: d/dx e^x = e^x. For any other base a > 0, d/dx a^x = a^x * ln(a). Notice that a^x is not a power function, so the power rule does not apply. Exponent and base play different roles.
Logarithms
The natural logarithm has derivative d/dx ln(x) = 1/x for x > 0. For a general base, d/dx log_a(x) = 1/(x ln(a)). A neat consequence of the chain rule: d/dx ln(g(x)) = g'(x)/g(x). So the derivative of ln(5x) is 5/(5x) = 1/x, the same as ln(x), because ln(5x) = ln(5) + ln(x) and ln(5) is just a constant.
Combining with earlier rules
Every earlier rule still works. Product, quotient and chain rules simply gain new building blocks.
Worked example
Differentiate w(x) = e^(2x) * sin(x).
- Product rule: w' = (e^(2x))' sin(x) + e^(2x) (sin(x))'.
- Chain rule for the first factor: (e^(2x))' = 2e^(2x).
- So w'(x) = 2e^(2x) sin(x) + e^(2x) cos(x) = e^(2x)(2 sin(x) + cos(x)).
A numeric check at x = 1 gives a slope of about 16.43 from both the formula and a central difference.
A second example: differentiate ln(x^2 + 1). The inside is x^2 + 1, with derivative 2x, so the result is 2x/(x^2 + 1).
A useful limit
Because the derivative of e^x at x = 0 equals 1, we get lim(h -> 0) (e^h - 1)/h = 1. With a scaling factor, (e^(kh) - 1)/h approaches k.
Common mistakes
- Dropping the minus sign in the derivative of cos(x).
- Applying the power rule to e^x and writing x*e^(x-1).
- Forgetting the chain-rule factor, such as differentiating sin(3x) as cos(3x).
- Thinking that d/dx ln(5x) is 5/x.