Trigonometric Derivatives

Trigonometric Derivatives, as with all derivatives, are used to find the rate of change of trigonometric functions. They are used in various real-world applications such as programming and electronics.

Below is a table outlining the derivatives of each trig function:

Function Original Derivative
Sine \(f(x) = \sin(x)\) \(f'(x) = \cos(x)\)
Cosine \(f(x) = \cos(x)\) \(f'(x) = -\sin(x)\)
Tangent \(f(x) = \tan(x)\) \(f'(x) = \cfrac{1}{\text{cos}^2x} = \sec^2(x)\)
Cotangent \(f(x) = \cot(x)\) \(f'(x) = -\cfrac{1}{\text{sin}^2x} = -\csc^2(x)\)
Secant \(f(x) = \sec(x)\) \(f(x)' = \tan(x)\sec(x)\)
Cosecant \(f'(x) = \csc(x)\) \(f'(x) = -\cot(x)\csc(x)\)

Example

Differentiate the following equations.

  1. \(y = \cos(3x)\)
  2. \(y = \tan(x)\sin(2x)\)

i. We can use Chain Rule to differentiate this equation:

\(y' = \textcolor{red}{f'(g(x))}\textcolor{blue}{g'(x)}\)

\(y' = \textcolor{red}{-\sin(3x)}\cdot \textcolor{blue}{3}\)

\(y' = -3\sin((3x))\)

Therefore, we can determine that \(y' = -3\sin(3x)\).


ii. We can use Product Rule to find the derivative of this expression:

First, we can identify \(f(x)\) and \(g(x)\):

\(\textcolor{red}{f(x) = \tan(x)}\)

\(\textcolor{blue}{g(x) = \sin(2x)}\)

We can determine \(f'(x)\) simply by differentiating \(f(x)\):

\(\textcolor{red}{f'(x) = \sec^2(x)}\)

We can determine \(g'(x)\) by using Chain Rule:

\(g'(x) = \cos(2x)\cdot 2\)

\(\textcolor{blue}{g'(x) = 2\cos(2x)}\)

Next, we can use Product Rule to differentiate the function:

\(y' = \textcolor{red}{f'(x)}\textcolor{blue}{g(x)} + \textcolor{blue}{g'(x)}\textcolor{red}{f(x)}\)

\(y' = \textcolor{red}{\sec^2(x)}\textcolor{blue}{\sin(2x)} + \textcolor{blue}{2\cos(2x)}\textcolor{red}{\tan(x)}\)

Therefore, we can determine that \(y' = \sec^2(x)\sin(2x) + 2\cos(2x)\tan(x)\).


Differentiate the following equations.

\(y = \cos^3(x^2+\pi x)\)


\(y = 2\sin^3(x)-4\cos^2(x)\)


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