$$\DeclareMathOperator{cosec}{cosec}$$

# Differentiation

## Furthermore

### Examples

1. Chain Rule:
The chain rule is used when differentiating composite functions. If we have a function $$y = f(u)$$ and $$u = g(x)$$, then the derivative of $$y$$ with respect to $$x$$ is given by:

$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

Example: Differentiate $$y = \sin(3x^2)$$ with respect to $$x$$.

Let $$u = 3x^2$$. Then, $$\frac{du}{dx} = 6x$$ and $$\frac{dy}{du} = \cos(u)$$. Using the chain rule:

$$\frac{dy}{dx} = \cos(3x^2) \cdot 6x = 6x \cos(3x^2)$$

2. Product Rule:
The product rule is used when differentiating the product of two functions. If $$y = u \cdot v$$ where both $$u$$ and $$v$$ are functions of $$x$$, then the derivative of $$y$$ with respect to $$x$$ is:

$$\frac{dy}{dx} = u' \cdot v + u \cdot v'$$

Example: Differentiate $$y = x^2 \cdot \ln(x)$$ with respect to $$x$$.

Using the product rule:

$$\frac{dy}{dx} = 2x \cdot \ln(x) + x^2 \cdot \frac{1}{x} = 2x \ln(x) + x$$

3. Quotient Rule:
The quotient rule is used when differentiating the quotient of two functions. If $$y = \frac{u}{v}$$ where both $$u$$ and $$v$$ are functions of $$x$$ and $$v \neq 0$$, then the derivative of $$y$$ with respect to $$x$$ is:

$$\frac{dy}{dx} = \frac{u' \cdot v - u \cdot v'}{v^2}$$

Example: Differentiate $$y = \frac{x^2}{\sin(x)}$$ with respect to $$x$$.

Using the quotient rule:

$$\frac{dy}{dx} = \frac{2x \cdot \sin(x) - x^2 \cdot \cos(x)}{\sin^2(x)}$$

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