Differentiate natural-logarithmic functions

Which first rule should be used for the displayed function?

Differentiating natural logarithms

The derivative of a natural logarithm depends on both the logarithm argument and how that argument changes. Write the inner derivative in the numerator and the original argument in the denominator

1. 1Use the basic log rule The derivative of the natural logarithm of x is the reciprocal of x. \(\displaystyle \frac{d}{dx}\log_e(x)=\frac{1}{x}\)
2. 2Keep the inner derivative For a logarithm of h(x), divide the derivative of h by h itself \(\displaystyle \frac{d}{dx}\log_e\left(h(x)\right)=\frac{h'(x)}{h(x)}\)
3. 3Carry constant multiples A constant multiplying a logarithm remains as a factor in the derivative \(\displaystyle \frac{d}{dx}\left[a\log_e\left(h(x)\right)\right]=a\frac{h'(x)}{h(x)}\)
4. 4Preserve the original domain The original logarithm argument must be positive, even when a log law gives a simpler derivative \(\displaystyle \log_e\left((x-2)^{2}\right)=2\log_e\left|x-2\right|,\quad x\ne2\)

Find the constant derivative of the displayed inner function

Find the exact gradient at x = 1. Enter a number, not a derivative rule

Omitting the inner derivative. One of these lines is right. Which one?

Composite arguments and their domains

Differentiate the whole inner function. Find the domain from its positive values, not from the simplified derivative

1. 1Polynomial argument The quadratic argument is positive only between its roots \(\displaystyle Q(x)=\log_e(9-x^2),\quad Q'(x)=\frac{-2x}{9-x^2},\quad -3<x<3\)
2. 2Circular argument The argument is at least 1, so every real x is allowed \(\displaystyle T(x)=\log_e(3+2\cos x),\quad T'(x)=\frac{-2\sin x}{3+2\cos x}\)
3. 3Exponential argument The argument is positive for every real x. The chain rule agrees with simplifying the original logarithm \(\displaystyle E(x)=\log_e(e^{3x})=3x,\quad E'(x)=3\)

Find the exact gradient at x = 2. Enter an integer or fraction

A product with a logarithm

Differentiate each factor in turn. Keep the original inner function in the rational denominator

1. 1Identify the factors Use the product rule, then the logarithmic chain rule \(\displaystyle P(x)=(x+1)\log_e(2x+1)\)
2. 2Keep both contributions The logarithmic term comes from differentiating the first factor \(\displaystyle P'(x)=\log_e(2x+1)+(x+1)\frac{2}{2x+1}\)
3. 3Read the coefficients Read the coefficients from the derivative and retain the original logarithm domain \(\displaystyle P'(x)=\log_e(2x+1)+\frac{2x+2}{2x+1},\quad A=1,\ B=2,\ C=2,\quad x>-\frac{1}{2}\)

Function

\(\displaystyle R(x)=(2x-1)\log_e(3x+2)\)

Differentiate R and find B in the stated form

A quotient with a logarithm

The quotient rule has a subtraction and a squared denominator. In this example the denominator is always positive, so only the logarithm restricts the domain

1. 1Name the parts The logarithmic argument is h and the denominator is u \(\displaystyle Q(x)=\frac{\log_e(2x+3)}{x^2+1},\quad h(x)=2x+3,\quad u(x)=x^2+1\)
2. 2Apply the quotient rule Differentiate the numerator with the chain rule \(\displaystyle Q'(x)=\frac{(2/h(x))u(x)-2x\log_e(h(x))}{u(x)^2}\)
3. 3Separate the terms The logarithmic numerator coefficient is negative two and the reciprocal numerator coefficient is two; retain the original logarithm domain \(\displaystyle Q'(x)=\frac{-2x\log_e(h(x))}{u(x)^2}+\frac{2}{h(x)u(x)},\quad x>-\frac{3}{2}\)

Function

\(\displaystyle F(x)=\frac{3\log_e(2x+1)}{x^2+4}\)

Named parts

\(\displaystyle h(x)=2x+1,\quad u(x)=x^2+4\)

Find A in the derivative form. Enter its signed numerical value

Rewriting a squared argument

The square allows either sign of the inner linear expression, except zero. The factor from the logarithm law and the original outer multiplier both remain

1. 1Preserve both sides The absolute value preserves the original domain \(\displaystyle H(x)=\log_e((2x-3)^2)=2\log_e|2x-3|,\quad x\ne\frac{3}{2}\)
2. 2Apply the chain rule Multiply the logarithm-law factor by the inner derivative \(\displaystyle H'(x)=2\frac{2}{2x-3}=\frac{4}{2x-3}\)
3. 3Keep the factors separate Keep both the original multiplier and the factor from the square in the derivative \(\displaystyle \frac{d}{dx}\left[A\log_e(h(x)^2)\right]=\frac{A P h'(x)}{h(x)},\quad P=2\)

Function

\(\displaystyle F(x)=-2\log_e((3x+1)^2)\)

Find A in the derivative form. Enter an exact number