# Differentiation of Trigonometric Functions

### Basic Trigonometric Derivatives:

1. Derivative of Sine Function:

• $\frac{d}{dx}\left[\mathrm{sin}\left(x\right)\right]=\mathrm{cos}\left(x\right)$
2. Derivative of Cosine Function:

• $\frac{d}{dx}\left[\mathrm{cos}\left(x\right)\right]=-\mathrm{sin}\left(x\right)$
3. Derivative of Tangent Function:

• $\frac{d}{dx}\left[\mathrm{tan}\left(x\right)\right]={\mathrm{sec}}^{2}\left(x\right)$
4. Derivative of Cosecant Function:

• $\frac{d}{dx}\left[\mathrm{csc}\left(x\right)\right]=-\mathrm{csc}\left(x\right)\cdot \mathrm{cot}\left(x\right)$
5. Derivative of Secant Function:

• $\frac{d}{dx}\left[\mathrm{sec}\left(x\right)\right]=\mathrm{sec}\left(x\right)\cdot \mathrm{tan}\left(x\right)$
6. Derivative of Cotangent Function:

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

### Using Trigonometric Identities:

1. Reciprocal Identities:

• $\mathrm{csc}\left(x\right)=\frac{1}{\mathrm{sin}\left(x\right)}$
• $\mathrm{sec}\left(x\right)=\frac{1}{\mathrm{cos}\left(x\right)}$
• $\mathrm{cot}\left(x\right)=\frac{1}{\mathrm{tan}\left(x\right)}$
2. Pythagorean Identities:

• ${\mathrm{sin}}^{2}\left(x\right)+{\mathrm{cos}}^{2}\left(x\right)=1$
• ${\mathrm{tan}}^{2}\left(x\right)+1={\mathrm{sec}}^{2}\left(x\right)$
• $1+{\mathrm{cot}}^{2}\left(x\right)={\mathrm{csc}}^{2}\left(x\right)$

### Derivatives of Inverse Trigonometric Functions:

1. Derivative of Arcsine Function:

• $\frac{d}{dx}\left[\mathrm{arcsin}\left(x\right)\right]=\frac{1}{\sqrt{1-{x}^{2}}}$
2. Derivative of Arccosine Function:

• $\frac{d}{dx}\left[\mathrm{arccos}\left(x\right)\right]=-\frac{1}{\sqrt{1-{x}^{2}}}$
3. Derivative of Arctangent Function:

• $\frac{d}{dx}\left[\mathrm{arctan}\left(x\right)\right]=\frac{1}{1+{x}^{2}}$
4. Derivative of Arcsecant Function:

• $\frac{d}{dx}\left[\text{arcsec}\left(x\right)\right]=\frac{1}{\mathrm{\mid }x\mathrm{\mid }\sqrt{{x}^{2}-1}}$

### Chain Rule with Trigonometric Functions:

When differentiating compositions involving trigonometric functions, the chain rule is applied. For instance:

• $\frac{d}{dx}\left[\mathrm{sin}\left(3{x}^{2}\right)\right]=\mathrm{cos}\left(3{x}^{2}\right)\cdot \frac{d}{dx}\left[3{x}^{2}\right]=3x\cdot \mathrm{cos}\left(3{x}^{2}\right)$