Derivatives · Definition
Derivatives of trigonometric functions
calculusderivativestrigonometry
Before you start you need to know
These derivatives are the foundation of almost every other trigonometric derivative in this course.
Definition
These are the derivatives of sine, cosine and tangent, with angles in radians; the others follow from the quotient and chain rules.
\[(\sin x)'=\cos x,\quad(\cos x)'=-\sin x,\quad(\tan x)'=\sec^{2}x\]
The derivatives of sine and cosine come from the notable limit \(\frac{\sin x}{x}\to1\): \((\sin x)'=\cos x\) and \((\cos x)'=-\sin x\). The minus sign of cosine is the one most often forgotten.
The others follow from the quotient rule: \((\tan x)'=\sec^2 x\), \((\sec x)'=\sec x\tan x\), etc.
All of them assume the angle is in radians; in degrees extra factors of \(\pi/180\) appear.
Think about it
What is the derivative of \(\cos x\)?
\(-\sin x\).
Exercises 10
Differentiate the following functions
- \(\displaystyle f(x)=\tan {x}\)
- \(\displaystyle f(x)=3 \sin {x} - 2 \cos {x}\)
- \(\displaystyle f(x)=\sec {x}\)
- \(\displaystyle f(x)=\cot {x}\)
- \(\displaystyle f(x)=\cos {x}\)
- \(\displaystyle f(x)=\sin {x}\)
- \(\displaystyle f(x)=x\sin {x}\)
- \(\displaystyle f(x)=\frac{\cos {x}}{x}\)
- \(\displaystyle f(x)=\sin {2x}\)
Problems
References
- James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.