Derivatives of the Trigonometric Functions

Sine, cosine, tangent, cosecant, secant, cotangent. These are functions that crop up continuously in mathematics and engineering and have a lot of practical applications. They also appear in more advanced mathematics, particularly when dealing with things such as line integrals with complex numbers and alternate representations of space like spherical and cylindrical coordinate systems.

We use the definition of the derivative, i.e.,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}} ,

to work these first two out.

Let us find the derivative of sin(x), using the above definition.

$f(x)=\sin(x)$
$f'(x)=\lim _{h\to 0}{\frac {\sin(x+h)-\sin(x)}{h}}$	Definition of derivative
$=\lim _{h\to 0}{\frac {\cos(x)\sin(h)+\cos(h)\sin(x)-\sin(x)}{h}}$	trigonometric identity
$=\lim _{h\to 0}{\frac {\cos(x)\sin(h)+(\cos(h)-1)\sin(x)}{h}}$	factoring
$=\lim _{h\to 0}{\frac {\cos(x)\sin(h)}{h}}+\lim _{h\to 0}{\frac {(\cos(h)-1)\sin(x)}{h}}$	separation of terms
$=\cos(x)\times 1+\sin(x)\times 0$	application of limit
$=\cos(x)$	solution

Now for the case of cos(x).

$f(x)=\cos(x)$
$f'(x)=\lim _{h\to 0}{\frac {\cos(x+h)-\cos(x)}{h}}$	Definition of derivative
$=\lim _{h\to 0}{\frac {\cos(x)\cos(h)-\sin(h)\sin(x)-\cos(x)}{h}}$	trigonometric identity
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\lim_{h\to 0}\frac{\cos(x)(\cos(h)-1)-\sin(x)\sin(h)}{h}}	factoring
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\lim_{h\to 0}\frac{\cos(x)(\cos(h)-1)}{h}-\lim_{h\to 0}\frac{\sin(x)\sin(h)}{h}}	separation of terms
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\cos(x)\times 0-\sin(x)\times 1}	application of limit
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =-\sin(x)}	solution

Therefore we have established

Derivative of Sine and Cosine Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}\sin(x)=\cos(x)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}\cos(x)=-\sin(x)}

To find the derivative of the tangent, we just remember that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan(x)=\frac{\sin(x)}{\cos(x)}}

which is a quotient. Applying the quotient rule, we get:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}\tan(x)=\frac{\cos^2(x)+\sin^2(x)}{\cos^2(x)}}

Then, remembering that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos^2(x)+\sin^2(x)=1} , we simplify:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\cos^2(x)+\sin^2(x)}{\cos^2(x)}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{1}{\cos^2(x)}}
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\sec^2(x)}

Derivative of the Tangent Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}\tan(x)=\sec^2(x)}

For secants, we again apply the quotient rule.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sec(x)=\frac{1}{\cos(x)}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align}\frac{d}{dx}\sec(x)&=\frac{d}{dx}\frac{1}{\cos(x)}\\ &=\frac{\cos(x)\frac{d 1}{dx}-1\frac{d\cos(x)}{dx}}{\cos(x)^2}\\ &=\frac{\cos(x)0-1(-\sin(x))}{\cos(x)^2} \end{align}}

Leaving us with:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}\sec(x)=\frac{\sin(x)}{\cos^2(x)}}

Simplifying, we get:

Derivative of the Secant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}\sec(x)=\sec(x)\tan(x)}

Using the same procedure on cosecants:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \csc(x)=\frac{1}{\sin(x)}}

We get:

Derivative of the Cosecant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}\csc(x)=-\csc(x)\cot(x)}

Using the same procedure for the cotangent that we used for the tangent, we get:

Derivative of the Cotangent Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}\cot(x)=-\csc^2(x)}

Resources

• The Derivatives of the Trigonometric Functions PowerPoint file created by Dr. Sara Shirinkam, UTSA.

• The Derivatives of the Trigonometric Functions Worksheet PowerPoint file created by Dr. Sara Shirinkam, UTSA.