Subject: Calculus

Derivatives Of Trigonometric Functions


So far, we have not encountered trigonometric functions in dealing with derivatives. Basically, derivatives of trigonometric functions are also trigonometric functions. Its basic structures, like any ordinary funcions, are also derivable from the three-step rule mentioned in the previous chapter in which we use the concept of limits. However, we will not go to that kinds of details. Instead, let me go directly to the derivatives of the six trigonometric functions.

Theorem (Derivatives of the six trigonometric functions)


1. D_{x}\sin{x}=\cos{x}

2. D_{x}\cos{x}=-\sin{x}

3. D_{x}\tan{x}=\sec^{2}{x}

4. D_{x}\cot{x}=-\csc^{2}{x}

5. D_{x}\sec{x}=\sec{x}\tan{x}

6. D_{x}\csc{x}=-\csc{x}\cot{x}

Example #1

Find the derivative of the following function.

f(x)=\cos{3x}

Solution:

f'(x)=D_{x}\cos{3x}
 =-\sin{3x} \cdot D_{x}{3x}
 =-3\sin{3x}.

Example #2

Find the derivative of the following function.

f(x)=\cos{3x}

Solution:

f'(x)=D_{x}\sec{ax}\sin{bx}
 =\sec{ax} \cdot D_{x}\sin{bx}
 =-3\sin{3x}.

Example #3

Differentiate the following trigonometric function.

f(x)=\tan(x^{3}-4)

Solution:

f'(x)=D_{x}\tan(x^{3}-4)
 =\sec^{2}(x^{3}-4) \cdot D_{x}(x^{3}-4)
 =3x^{2} \sec^{2}(x^{3}-4).

Example #4

Differentiate the following trigonometric function.

f(x)=\cos{\sqrt{5x}}

Solution:

f'(x)=D_{x}\cos{\sqrt{5x}}
 =-\sin{\sqrt{5x}} \cdot D_{x}{5x}
 =-5\sin{\sqrt{5x}}.

Example #5

Differentiate the following trigonometric function.

f(x)=\sin{\cos{5x}}

Solution:

f'(x)=D_{x}\sin{\cos{5x}}
 =\cos{\cos{5x}} \cdot D_{x}\cdot \cos{5x}
 =-5\cos{\cos{5x}}\cdot \sin{5x}.

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