Basic Trigonometric Functions:

• Sine: $\frac{d}{dx} \sin(x) = \cos(x)$
• Cosine: $\frac{d}{dx} \cos(x) = -\sin(x)$
• Tangent: $\frac{d}{dx} \tan(x) = \sec^2(x)$
• Cosecant: $\frac{d}{dx} \csc(x) = -\csc(x) \cot(x)$
• Secant: $\frac{d}{dx} \sec(x) = \sec(x) \tan(x)$
• Cotangent: $\frac{d}{dx} \cot(x) = -\csc^2(x)$

Natural Logarithm ($ln$):

• Logarithm of $x$: $\frac{d}{dx} \ln(x) = \frac{1}{x}$
• Logarithm of a function $u(x)$: $\frac{d}{dx} \ln(u(x)) = \frac{1}{u(x)} \cdot u'(x)$
• This is often used in combination with the chain rule.

Logarithm with a Different Base:

• Logarithm base $\ a$: $\frac{d}{dx} \log_a(x) = \frac{1}{x \ln(a)}$
• The derivative of $\ \log_a(x)$ depends on the base $\ a$ and involves the natural logarithm $\ \ln(a) .$

General Exponential Function:

• Exponential base $\ a$: $\frac{d}{dx} a^x = a^x \ln(a)$
• Exponential of a function $\ u(x)$: $\frac{d}{dx} a^{u(x)} = a^{u(x)} \cdot \ln(a) \cdot u'(x)$

Natural Exponential Function:

• Exponential of $\ x$: $\frac{d}{dx} e^x = e^x$
• Exponential of a function $\ u(x)$: $\frac{d}{dx} e^{u(x)} = e^{u(x)} \cdot u'(x)$
• This is also an application of the chain rule.

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

• Apply the chain rule: $\frac{dy}{dx} = \cos(3x) \cdot 3 = 3\cos(3x)$

Example 2: Differentiate $\ y = \ln(2x + 1)$

• Again, use the chain rule: $\frac{dy}{dx} = \frac{1}{2x + 1} \cdot 2 = \frac{2}{2x + 1}$

Example 3: Differentiate $\ y = e^{4x^2}$

• Use the chain rule: $\frac{dy}{dx} = e^{4x^2} \cdot 8x$

Example 4: Differentiate $\ y = \tan(\ln(x))$

• Combine the chain rule with the derivative of the tangent function: $\frac{dy}{dx} = \sec^2(\ln(x)) \cdot \frac{1}{x}$

• Differentiating trigonometric functions, logarithmic functions, and exponential functions is straightforward once you know the basic rules and how to apply the chain rule.
• The chain rule is particularly important for handling composite functions.
• Mastering these techniques allows you to solve complex problems in mathematics, physics, economics, and other fields that involve rates of change.