Differentiation Techniques Notes — JC H2

Logarithm of sec x + tan x

$\frac{d}{dx}\ln(\sec x + \tan x) = \sec x$

Chain rule on a logarithm; the result simplifies to a single term.

Logarithm of sin²x

$\frac{d}{dx}\ln(\sin^2 x) = 2\cot x$

Chain rule; the answer is written in terms of cot x.

Inverse sine with a scaled argument

$\frac{d}{dx}\sin^{-1}\left(\frac{x}{10}\right) = \frac{1}{\sqrt{100 - x^2}}$

Inverse trigonometric function of x ÷ 10.

Product of exponential and trigonometric functions

$\frac{d}{dx}\left[e^{3x}\sin^2 2x\right] = 3e^{3x}\sin^2 2x + 4e^{3x}\sin 2x\cos 2x$

Product rule, with the chain rule for e^(3x) and for sin²2x.

Exponential with base 5

$\frac{d}{dx}\left(5^{\sqrt{x}}\right) = \frac{5^{\sqrt{x}}\ln 5}{2\sqrt{x}}$

Base 5 with exponent √x; the factor ln 5 comes from the base.

Variable base and power

$\frac{d}{dx}\left[(x^2+x)^x\right] = (x^2+x)^x\left[\frac{2x+1}{x+1} + \ln(x^2+x)\right]$

Both the base and the exponent contain x.