Calculus is fundamental for analyzing change, and differentiation is one of its primary operations. Below, we'll cover some essential differentiation rules, including the Power Rule, Product Rule, Quotient Rule, and trigonometric derivatives.

2.5 Applying the Power Rule

The Power Rule simplifies finding derivatives for functions of the form \( f(x) = x^n \), where \( n \) is any real number.

Power Rule Formula

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

For example, to find the derivative of \( x^3 \), we apply the Power Rule as follows:

\[ \frac{d}{dx} x^3 = 3 \cdot x^{3-1} = 3x^2 \]

2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple

These basic rules help when working with functions that involve constants, sums, or products.

1. Constant Rule

The derivative of a constant is zero. If \( f(x) = c \), then \( f'(x) = 0 \).

2. Sum Rule

The derivative of a sum of functions is the sum of their derivatives:

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

3. Difference Rule

The derivative of a difference of functions is the difference of their derivatives:

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

4. Constant Multiple Rule

The derivative of a constant times a function is the constant times the derivative of the function:

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

2.7 Trigonometric Derivatives: Sine, Cosine, Exponential, and Logarithmic Functions

Understanding the derivatives of basic trigonometric, exponential, and logarithmic functions is fundamental in calculus. Here are the primary derivatives:

1. Derivative of \( \sin x \)

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

2. Derivative of \( \cos x \)

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

3. Derivative of \( a^x \)   and    \( e^x\)

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

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

4. Derivative of \( \ln x \)   and   \(\log _a x \)

\[ \frac{d}{dx} \log_a x = \frac{1}{x} \cdot \frac{1}{\ln a} \]

\[ \frac{d}{dx} \ln x = \frac{d}{dx} \log_e x = \frac{1}{x} \cdot \frac{1}{\ln e} = \frac{1}{x} \]

2.8 The Product Rule

The Product Rule is essential for differentiating products of two functions. If \( h(x) = f(x) \cdot g(x) \), then:

\[ h'(x) = f(x) \cdot g'(x) + g(x) \cdot f'(x) \]

This rule is useful when dealing with expressions where two functions are multiplied together, such as \( h(x) = x^2 \sin x \).

2.9 The Quotient Rule

The Quotient Rule is used for differentiating a quotient of two functions. If \( h(x) = \frac{f(x)}{g(x)} \), then:

\[ h'(x) = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2} \]

For example, to differentiate \( h(x) = \frac{x^3}{\cos x} \), we apply the Quotient Rule.

2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and Cosecant Functions

In addition to sine and cosine, the derivatives of tangent, cotangent, secant, and cosecant functions are also frequently used in calculus:

1. Derivative of \( \tan x \)

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

2. Derivative of \( \cot x \)

\[ \frac{d}{dx} (\cot x) = -\csc^2 x \]

3. Derivative of \( \sec x \)

\[ \frac{d}{dx} (\sec x) = \sec x \cdot \tan x \]

4. Derivative of \( \csc x \)

\[ \frac{d}{dx} (\csc x) = -\csc x \cdot \cot x \]