Derivative Rules

1. Constant Rule

The derivative of a constant is zero.

$$\frac{d}{dx}[c] = 0$$

2. Power Rule

For a function of the form $f(x) = x^n$, the derivative is:

$$\frac{d}{dx}[x^n] = n x^{n-1}$$

3. Sum Rule

The derivative of a sum of two functions is the sum of their derivatives.

$$\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$$

4. Difference Rule

The derivative of a difference is the difference of the derivatives.

$$\frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x)$$

5. Product Rule

For the product of two functions, the derivative is:

$$\frac{d}{dx}[f(x) g(x)] = f'(x) g(x) + f(x) g'(x)$$

6. Quotient Rule

The derivative of a quotient of two functions is:

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x) g(x) - f(x) g'(x)}{[g(x)]^2}$$

7. Chain Rule

The derivative of a composite function is:

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

8. Exponential Functions

The derivative of exponential functions is:

$$\frac{d}{dx}[e^x] = e^x$$

$$\frac{d}{dx}[a^x] = a^x \ln(a)$$

9. Logarithmic Functions

The derivative of logarithmic functions is:

$$\frac{d}{dx}[\ln(x)] = \frac{1}{x}$$

$$\frac{d}{dx}[\log_a(x)] = \frac{1}{x \ln(a)}$$

10. Trigonometric Functions

Derivatives of the basic trigonometric functions:

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

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

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

11. Inverse Trigonometric Functions

Derivatives of inverse trigonometric functions:

$$\frac{d}{dx}[\arcsin(x)] = \frac{1}{\sqrt{1 - x^2}}, \quad |x| < 1$$

$$\frac{d}{dx}[\arctan(x)] = \frac{1}{1 + x^2}$$

12. Hyperbolic Functions

Derivatives of hyperbolic sine and cosine:

$$\frac{d}{dx}[\sinh(x)] = \cosh(x)$$

$$\frac{d}{dx}[\cosh(x)] = \sinh(x)$$

$$\frac{d}{dx}[\tanh(x)] = \operatorname{sech}^2(x)$$