Logarithmic differentiation of y = f(x)

How it works

For y = f(x), logarithmic differentiation takes the natural logarithm of both sides, expands it with the logarithm laws, and differentiates:

ln|y| = ln|f(x)|, then y′/y = d/dx ln|f(x)|, so dy/dx = y · (y′/y)

The page expands ln y from the outside of f in:

• ln|u v| = ln|u| + ln|v| and ln|u/v| = ln|u| − ln|v|;
• ln|u^v| = v ln|u|, and ln(e^v) = v, ln(exp(v)) = v;
• ln|√u| = ln|u|/2;
• ln|−u| = ln|u|;
• any other part u stays as ln|u| (a number c as ln(c), without bars when it is positive; the constants e and π without bars).

A computer algebra system (nerdamer, open source) then differentiates ln y; the algebra runs in the background after you start typing. The page shows three things:

• ln y = the expanded logarithm, with bars.
• y′/y = its derivative, as the algebra writes it (without bars, since the derivative of ln|u| is u′/u, the same as that of ln u).
• dy/dx = y times y′/y, written as f, a space, and y′/y in brackets: x^x (1 + ln(x)). When f as written starts with a minus sign or has a + or − between terms anywhere (even inside brackets), f is put in brackets too: (x^2 + 1) (2x/(1 + x^2)).

Every answer is checked before it is shown. The derivative of ln y is compared with a numeric difference quotient at 20 points (to 1 part in a million). The expanded ln y is compared with ln|f(x)| at 18 test points (−33.1, −7.31, −3.17, −1.73, −0.91, −0.37, −0.043, 0.21, 0.57, 0.77, 1.33, 2.63, 3.37, 4.19, 8.93, 17.3, 41.3 and 97.7): at each point where both are real numbers they must agree to 1 part in 10⁹ of the larger size (at least 1), and there must be at least 3 such points. If a check fails, the algebra finds no formula, or the work takes over 3 seconds, the page says "No verified answer". A result that holds a number the algebra could only give rounded (a whole number past 2⁵³; a fraction p/q, p and q being the numbers that multiply its top and its bottom, such as 288557167/(342919925e), when q after removing its factors 2 and 5 is over 1,000,000, or is over 1 while |p| times it is over 10¹²; or a decimal with more than 12 significant digits) is not shown.

What you can type

• A number has at most 15 digits in a row. A computer number keeps only about 16 digits, so a longer one (9007199254740993) would stand for a nearby number (9007199254740992), and the page asks for fewer digits instead. Write very large or very small numbers with a power of ten (1e-20).
• y = f(x) uses the variable x. Numbers can have decimals (2.5) and powers of ten (1e-3).
• Operations: + − * / and ^ for powers. Brackets group. A number or bracket next to a letter multiplies: 2x, 3(x + 1), x sin(x).
• Constants: pi (or π) and e.
• Functions: sqrt, cbrt, ln (and log, the same natural logarithm), log10, exp, abs (or |x|), sin, cos, tan, sec, csc, cot, asin, acos, atan (arcsin, arccos, arctan also work), sinh, cosh, tanh and their inverses. Angles are in radians. sin x without brackets means sin(x); sin x^2 means sin(x²).

Worked examples by hand

y = (2x⁴ + 1)^tan(x) (OpenStax Calculus Volume 1, section 3.9, Example 3.81). ln y = tan(x) ln|2x⁴ + 1|. By the product rule, y′/y = sec²(x) ln(2x⁴ + 1) + tan(x) · 8x³/(2x⁴ + 1), which the page writes 8x^3 tan(x)/(1 + 2x^4) + ln(1 + 2x^4) sec(x)^2. So dy/dx = (2x⁴ + 1)^tan(x) (8x³ tan(x)/(1 + 2x⁴) + ln(1 + 2x⁴) sec²(x)).

y = x √(2x + 1)/(eˣ sin³x) (OpenStax Calculus Volume 1, section 3.9, Example 3.82). ln y = ln|x| + ln|2x + 1|/2 − x − 3 ln|sin(x)|. Differentiating term by term, y′/y = 1/x + 1/(2x + 1) − 1 − 3 cot(x), which the page writes 1/(1 + 2x) − 1 − 3 cos(x)/sin(x) + 1/x.