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Derivative Formulas

1. **Constant Rule**:

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


2. **Power Rule**:
\[
\frac{d}{dx} [x^n] = nx^{n-1}
\]

3. **Sum Rule**:
\[
\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)
\]

4. **Difference Rule**:
\[
\frac{d}{dx} [f(x) - g(x)] = f'(x) - g'(x)
\]

5. **Product Rule**:
\[
\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
\]

6. **Quotient Rule**:
\[
\frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}
\]

7. **Chain Rule**:
\[
\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)
\]

8. **Exponential Function**:
\[
\frac{d}{dx} [e^x] = e^x
\]

9. **Natural Logarithm**:
\[
\frac{d}{dx} [\ln x] = \frac{1}{x}
\]

10. **Trigonometric Functions**:
\[
\frac{d}{dx} [\sin x] = \cos x
\]
\[
\frac{d}{dx} [\cos x] = -\sin x
\]
\[
\frac{d}{dx} [\tan x] = \sec^2 x
\]

Integration Formulas

1. **Constant Rule**:
\[
\int c \, dx = cx + C
\]

2. **Power Rule**:
\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)
\]

3. **Sum Rule**:
\[
\int [f(x) + g(x)] \, dx = \int f(x) \, dx + \int g(x) \, dx
\]

4. **Difference Rule**:
\[
\int [f(x) - g(x)] \, dx = \int f(x) \, dx - \int g(x) \, dx
\]

5. **Exponential Function**:
\[
\int e^x \, dx = e^x + C
\]

6. **Natural Logarithm**:
\[
\int \frac{1}{x} \, dx = \ln |x| + C
\]

7. **Trigonometric Functions**:
\[
\int \sin x \, dx = -\cos x + C
\]
\[
\int \cos x \, dx = \sin x + C
\]
\[
\int \sec^2 x \, dx = \tan x + C
\]

8. **Integration by Parts**:
\[
\int u \, dv = uv - \int v \, du
\]