Derivatives are a fundamental concept in calculus that measure the rate of change of a function with respect to a variable. While the theory behind derivatives can be complex, applying the basic rules of differentiation can be straightforward once you understand them. ## The Fundamental Rules of Differentiation Here are the core rules that form the foundation of differentiation: ### 1. The Power Rule For any real number n, the derivative of $x^n$ is: $$\frac{d}{dx}[x^n] = nx^{n-1}$$ Example: $\frac{d}{dx}[x^3] = 3x^2$ ### 2. The Constant Rule The derivative of a constant is zero:

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

Example: $\frac{d}{dx}[5] = 0$ ### 3. The Sum Rule The derivative of a sum is the sum of the derivatives:

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

Example: $\frac{d}{dx}[x^2 + 3x] = \frac{d}{dx}[x^2] + \frac{d}{dx}[3x] = 2x + 3$ ### 4. The Product Rule For the product of two functions:

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

Example: $\frac{d}{dx}[x^2 \cdot \sin(x)] = 2x \cdot \sin(x) + x^2 \cdot \cos(x)$ ### 5. The Quotient Rule For the quotient of two functions:

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

Example: $\frac{d}{dx}\left[\frac{x^2}{\cos(x)}\right] = \frac{2x \cdot \cos(x) - x^2 \cdot (-\sin(x))}{[\cos(x)]^2} = \frac{2x\cos(x) + x^2\sin(x)}{\cos^2(x)}$ ### 6. The Chain Rule For a composite function $f(g(x))$:

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

Example: $\frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x = 2x\cos(x^2)$ ## Common Derivatives to Memorize These derivatives appear frequently in calculus problems: - $\frac{d}{dx}[\sin(x)] = \cos(x)$ - $\frac{d}{dx}[\cos(x)] = -\sin(x)$ - $\frac{d}{dx}[\tan(x)] = \sec^2(x)$ - $\frac{d}{dx}[e^x] = e^x$ - $\frac{d}{dx}[\ln(x)] = \frac{1}{x}$ ## Applying Derivative Rules: A Complete Example Let's find the derivative of $f(x) = x^3\sin(x^2) + \frac{e^x}{\ln(x)}$. We'll break this down step by step:

\begin{align} f'(x) &= \frac{d}{dx}[x^3\sin(x^2)] + \frac{d}{dx}\left[\frac{e^x}{\ln(x)}\right] \\ \end{align}

For the first term, we use the product rule:

\begin{align} \frac{d}{dx}[x^3\sin(x^2)] &= \frac{d}{dx}[x^3] \cdot \sin(x^2) + x^3 \cdot \frac{d}{dx}[\sin(x^2)] \\ &= 3x^2 \cdot \sin(x^2) + x^3 \cdot \cos(x^2) \cdot \frac{d}{dx}[x^2] \\ &= 3x^2 \cdot \sin(x^2) + x^3 \cdot \cos(x^2) \cdot 2x \\ &= 3x^2\sin(x^2) + 2x^4\cos(x^2) \end{align}

For the second term, we use the quotient rule:

\begin{align} \frac{d}{dx}\left[\frac{e^x}{\ln(x)}\right] &= \frac{\frac{d}{dx}[e^x] \cdot \ln(x) - e^x \cdot \frac{d}{dx}[\ln(x)]}{[\ln(x)]^2} \\ &= \frac{e^x \cdot \ln(x) - e^x \cdot \frac{1}{x}}{[\ln(x)]^2} \\ &= \frac{e^x\ln(x) - \frac{e^x}{x}}{[\ln(x)]^2} \\ &= e^x\frac{x\ln(x) - 1}{x[\ln(x)]^2} \end{align}

Combining the results:

f'(x) = 3x^2\sin(x^2) + 2x^4\cos(x^2) + e^x\frac{x\ln(x) - 1}{x[\ln(x)]^2}

## Practice Makes Perfect Understanding these rules is one thing, but mastery comes from practice. Try working through various problems, starting with simple functions and gradually moving to more complex ones. Remember that differentiation is a skill that improves with practice. Each problem you solve builds your intuition and makes the next one easier. Don't be discouraged by complex expressions—break them down using the rules above, and you'll find that most derivatives become manageable.

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Derivative Rules Simplified: Mastering the Basics