Derivative of Trigonometric Functions

Trigonometric functions are fundamental in mathematics, and understanding their derivatives is essential for solving problems in calculus, physics, and engineering. In this in-depth guide, we will explore the derivatives of the six basic trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant.

Basic Derivatives of Trigonometric Functions

The derivatives of the basic trigonometric functions can be derived using the limit definition of the derivative and trigonometric identities. Below is a table summarizing these derivatives:

Function	Derivative
$\sin(x)$	$\cos(x)$
$\cos(x)$	$-\sin(x)$
$\tan(x)$	$\sec^2(x)$
$\cot(x)$	$-\csc^2(x)$
$\sec(x)$	$\sec(x)\tan(x)$
$\csc(x)$	$-\csc(x)\cot(x)$

Formulas and Examples

Derivative of Sine Function

The derivative of the sine function is the cosine function:

$$ \frac{d}{dx}[\sin(x)] = \cos(x) $$

Example:

Find the derivative of $f(x) = \sin(2x)$.

Using the chain rule:

$$ f'(x) = \cos(2x) \cdot \frac{d}{dx}[2x] = 2\cos(2x) $$

Derivative of Cosine Function

The derivative of the cosine function is the negative sine function:

$$ \frac{d}{dx}[\cos(x)] = -\sin(x) $$

Example:

Find the derivative of $f(x) = \cos(3x + 2)$.

Using the chain rule:

$$ f'(x) = -\sin(3x + 2) \cdot \frac{d}{dx}[3x + 2] = -3\sin(3x + 2) $$

Derivative of Tangent Function

The derivative of the tangent function is the square of the secant function:

$$ \frac{d}{dx}[\tan(x)] = \sec^2(x) $$

Example:

Find the derivative of $f(x) = \tan(x^2)$.

Using the chain rule:

$$ f'(x) = \sec^2(x^2) \cdot \frac{d}{dx}[x^2] = 2x\sec^2(x^2) $$

Derivative of Cotangent Function

The derivative of the cotangent function is the negative square of the cosecant function:

$$ \frac{d}{dx}[\cot(x)] = -\csc^2(x) $$

Example:

Find the derivative of $f(x) = \cot(\sqrt{x})$.

Using the chain rule:

$$ f'(x) = -\csc^2(\sqrt{x}) \cdot \frac{d}{dx}[\sqrt{x}] = -\frac{1}{2}x^{-\frac{1}{2}}\csc^2(\sqrt{x}) $$

Derivative of Secant Function

The derivative of the secant function is the product of the secant function and the tangent function:

$$ \frac{d}{dx}[\sec(x)] = \sec(x)\tan(x) $$

Example:

Find the derivative of $f(x) = \sec(e^x)$.

Using the chain rule:

$$ f'(x) = \sec(e^x)\tan(e^x) \cdot \frac{d}{dx}[e^x] = e^x\sec(e^x)\tan(e^x) $$

Derivative of Cosecant Function

The derivative of the cosecant function is the negative product of the cosecant function and the cotangent function:

$$ \frac{d}{dx}[\csc(x)] = -\csc(x)\cot(x) $$

Example:

Find the derivative of $f(x) = \csc(5x)$.

Using the chain rule:

$$ f'(x) = -\csc(5x)\cot(5x) \cdot \frac{d}{dx}[5x] = -5\csc(5x)\cot(5x) $$

Important Points to Remember

• The derivatives of trigonometric functions are periodic, just like the functions themselves.
• The chain rule is often used when differentiating composite functions involving trigonometric functions.
• It's important to be familiar with trigonometric identities, as they can simplify the differentiation process.
• The signs of the derivatives (positive or negative) are consistent with the behavior of the original functions over their respective domains.

By understanding these derivatives and practicing their application, you will be well-prepared to tackle a wide range of problems involving trigonometric functions in calculus.