What are the various rules involved when differentiating or integrating . How do we integrate or differentiate trigonometric functions like sinO

Okay, Ismaella. Let's look at the common rules for differentiation and integration, including trigonometric functions.

Differentiation Rules

Here are the main rules you'll use:

1. Constant Rule: The derivative of a constant is zero.

   $$\frac{d}{dx}(c) = 0$$

   Example: $\frac{d}{dx}(5) = 0$

2. Power Rule: (As discussed before) If $y = x^n$, then $\frac{dy}{dx} = nx^{n-1}$. Example: $\frac{d}{dx}(x^3) = 3x^2$

3. Constant Multiple Rule: If $y = cf(x)$, where $c$ is a constant, then $\frac{dy}{dx} = c \frac{d}{dx}(f(x))$. Example: $\frac{d}{dx}(4x^2) = 4 \frac{d}{dx}(x^2) = 4(2x) = 8x$

4. Sum/Difference Rule: If $y = f(x) \pm g(x)$, then $\frac{dy}{dx} = \frac{d}{dx}(f(x)) \pm \frac{d}{dx}(g(x))$. You differentiate each term separately. Example: $\frac{d}{dx}(x^2 + 3x) = \frac{d}{dx}(x^2) + \frac{d}{dx}(3x) = 2x + 3$

5. Trigonometric Functions:

   • $\frac{d}{dx}(\sin x) = \cos x$
   • $\frac{d}{dx}(\cos x) = -\sin x$
   • $\frac{d}{dx}(\tan x) = \sec^2 x$

6. Chain Rule: Used when you have a function inside another function, like $y = f(g(x))$.

   $$\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$$

   Example: If $y = \sin(2x)$, let $u = 2x$. Then $y = \sin u$. $\frac{dy}{du} = \cos u$ and $\frac{du}{dx} = 2$. So, $\frac{dy}{dx} = \cos(2x) \cdot 2 = 2\cos(2x)$.

Integration Rules

Integration rules often mirror differentiation rules in reverse. Remember to add the constant of integration, $C$.

1. Constant Multiple Rule: $\int c f(x)\,dx = c \int f(x)\,dx$ Example: $\int 4x^2\,dx = 4 \int x^2\,dx = 4 \left(\frac{x^3}{3}\right) + C = \frac{4}{3}x^3 + C$

2. Sum/Difference Rule: $\int (f(x) \pm g(x))\,dx = \int f(x)\,dx \pm \int g(x)\,dx$. You integrate each term separately. Example: $\int (x^2 + 3x)\,dx = \int x^2\,dx + \int 3x\,dx = \frac{x^3}{3} + \frac{3x^2}{2} + C$

3. Power Rule: (As discussed before) $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$). Example: $\int x^3\,dx = \frac{x^4}{4} + C$

4. Trigonometric Functions:

   • $\int \cos x\,dx = \sin x + C$
   • $\int \sin x\,dx = -\cos x + C$
   • $\int \sec^2 x\,dx = \tan x + C$

Examples with Trigonometric Functions

Example 1: Differentiate $y = 5\cos x - \sin(3x)$