Derivatives of Trigonometric Functions
Introduction
Trigonometric functions are fundamental in calculus because they appear frequently in physics, engineering, and geometry. Understanding how to differentiate them allows us to analyze oscillations, wave motion, and circular motion.
In this lecture, you will learn how to find derivatives of the six basic trigonometric functions.
Derivative Formulas
Let’s go over each one carefully.
Derivative of Sine
$$\frac{d}{dx}[sin(x)]=cos(x)$$
$$\frac{d}{dx}[sin(u)]=cos(u)\frac{du}{dx}$$
$$\lim\limits_{h \to 0}\frac{sin(x+h)-sin(x)}{h}$$
$$=\lim\limits_{h \to 0}\frac{sin(x)cos(h)+sin(h)cos(x)-sin(x)}{h}$$
$$=\lim\limits_{h \to 0}\frac{sin(x)(cos(h)-1)+sin(h)cos(x)}{h}$$
$$=\lim\limits_{h \to 0}\frac{sin(x)(cos(h)-1)}{h}+\lim\limits_{h \to 0}\frac{sin(h)cos(x)}{h}$$
$$=sin(x)\cdot{0}+1\cdot cos(x)$$
$$=cos(x)$$
example)
$$g(x)=sin(2x)$$
$$g^\prime(x)=cos(2x)\cdot(2x)^\prime$$
$$=cos(2x)\cdot2=2cos(2x)$$
$$h(x)=sin(x^3)$$
$$h^\prime(x)=cos(x^3)\cdot(x^3)^\prime$$
$$=cos(x^3)\cdot(3x^2)=3x^2cos(x^3)$$
Derivative of Cosine
$$\frac{d}{dx}[cos(x)]=-sin(x)$$
$$\frac{d}{dx}[cos(u)]=-sin(u)\frac{du}{dx}$$
$$\lim\limits_{h \to 0}\frac{cos(x+h)-cos(x)}{h}$$
$$=\lim\limits_{h \to 0}\frac{cos(x)cos(h)-sin(h)sin(x)-cos(x)}{h}$$
$$=\lim\limits_{h \to 0}\frac{cos(x)(cos(h)-1)-sin(h)sin(x)}{h}$$
$$=\lim\limits_{h \to 0}\frac{cos(x)(cos(h)-1)}{h}-\lim\limits_{h \to 0}\frac{sin(h)sin(x)}{h}$$
$$=cos(x)\cdot{0}-1\cdot sin(x)$$
$$=-sin(x)$$
example)
$$g(x)=2xcos(x^2)$$
$$g^\prime(x)=(2x)^\prime\cdot cos(x^2)+2x\cdot(cos(x^2))^\prime$$
$$=2\cdot cos(x^2)+2x\cdot -sin(x^2)\cdot(x^2)^\prime$$
$$=2cos(x^2)+2x\cdot -sin(x^2)\cdot(2x)$$
$$=2cos(x^2)-4x^2sin(x^2)$$
Derivative of Tangent
$$\frac{d}{dx}[tan(x)]=sec^2(x)$$
$$\frac{d}{dx}[tan(u)]=sec^2(u)\frac{du}{dx}$$
$$f(x)=tan(x)=\frac{sin(x)}{cos(x)}$$
$$f^\prime(x)=\frac{(sin(x))^\prime\cdot cos(x)-sin(x)(cos(x))^\prime}{(cos(x))^2}$$
$$=\frac{cos(x)\cdot cos(x)-sin(x)\cdot(-sin(x))}{cos^2(x)}$$
$$=\frac{cos^2(x)+sin^2(x)}{cos^2(x)}$$
$$=\frac{1}{cos^2(x)}$$
$$=sec^2(x)$$
example)
$$g(x)=\frac{tan(x)}{sin(x)}$$
$$g^\prime=\frac{(tan(x))^\prime\cdot sin(x)-tan(x)\cdot(sin(x))^\prime}{(sin(x))^2}$$
$$=\frac{sec^2(x)sin(x)-tan(x)cos(x)}{cos^2(x)}$$
Derivative of Cotangent
$$\frac{d}{dx}[cot(x)]=-csc^2(x)$$
$$\frac{d}{dx}[cot(u)]=-csc^2(u)\frac{du}{dx}$$
$$f(x)=cot(x)=\frac{cos(x)}{sin(x)}$$
$$f^\prime(x)=\frac{(cos(x))^\prime\cdot sin(x)-cos(x)(sin(x))^\prime}{(sin(x))^2}$$
$$=\frac{-sin(x)\cdot sin(x)-cos(x)\cdot cos(x)}{sin^2(x)}=\frac{-(sin^2(x)+cos^2(x))}{sin^2(x)}$$
$$=\frac{-1}{sin^2(x)}=-csc^2(x)$$
example)
$$g(x)=cot(3x^2)$$
$$g^\prime(x)=-csc(3x^2)cot(3x^2)\cdot(3x^2)^\prime$$
$$=-6x csc(3x^2)cot(3x^2)$$
Derivative of Secant
$$\frac{d}{dx}[sec(x)]=sec(x)tan(x)$$
$$\frac{d}{dx}[sec(u)]=sec(u)tan(u)\frac{du}{dx}$$
$$f(x)=sex(x)=\frac{1}{cos(x)}$$
$$f^\prime(x)=\frac{1^\prime\cdot cos(x)-1\cdot(cos(x))^\prime}{(cos(x))^2}$$
$$=\frac{sin(x)}{cos^2(x)}=\frac{1}{cos(x)}\cdot\frac{sin(x)}{cos(x)}$$
$$=sec(x)tan(x)$$
example)
$$g(x)=sec(4x)$$
$$g^\prime(x)=sec(4x)tan(4x)\cdot(4x)^\prime$$
$$=4sec(4x)tan(4x)$$
Derivative of Cosecant
$$\frac{d}{dx}[csc(u)]=-csc(u)cot(u)\frac{du}{dx}$$
$$f(x)=csc(x)=\frac{1}{sin(x)}$$
$$f^\prime(x)=\frac{1^\prime\cdot(sin(x))-1\cdot(sin(x))^\prime}{(sin(x))^2}$$
$$=\frac{-cos(x)}{sin^2(x)}=\frac{1}{sin(x)}\cdot\frac{-cos(x)}{sin(x)}$$
$$=-csc(x)cot(x)$$
example)
$$g(x)=csc(x^2+4)$$
$$g^\prime(x)=-csc(x^2+4)cot(x^2+4)\cdot(x^2+4)^\prime$$
$$=-2xcsc(x^2+4)cot(x^2+4)$$