# Nakafa Framework: LLM

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---

export const metadata = {
    title: "Derivative of Trigonometric Function",
    description: "Master derivatives of trigonometric functions: sin, cos, tan, and more. Learn formulas, product rule, quotient rule with step-by-step examples and solutions.",
    authors: [{ name: "Nabil Akbarazzima Fatih" }],
    date: "05/26/2025",
    subject: "Derivative Functions",
};

## The Rules of Trigonometric Functions

Finding the derivative of a trigonometric function is not much different from an algebraic function. We still use the familiar derivative properties, such as the product and quotient rules.

However, before that, we need to know the basic derivatives of the main trigonometric functions like sine and cosine. These basic rules are the foundation for solving more complex derivatives.

## Basic Trigonometric Derivatives

Just like the derivative of a power function, the derivative for a trigonometric function also has its basic pattern. The derivatives for sine and cosine, for example, can be proven directly from the limit definition of a derivative.

Here are the basic derivatives of the six trigonometric functions that we need to memorize:

1.  **Derivative of Sine**:
    
    <BlockMath math="f(x) = \sin x \implies f'(x) = \cos x" />

2.  **Derivative of Cosine**:

    <BlockMath math="f(x) = \cos x \implies f'(x) = -\sin x" />

3.  **Derivative of Tangent**:

    <BlockMath math="f(x) = \tan x \implies f'(x) = \sec^2 x" />

4.  **Derivative of Cotangent**:

    <BlockMath math="f(x) = \cot x \implies f'(x) = -\csc^2 x" />

5.  **Derivative of Secant**:

    <BlockMath math="f(x) = \sec x \implies f'(x) = \sec x \tan x" />

6.  **Derivative of Cosecant**:

    <BlockMath math="f(x) = \csc x \implies f'(x) = -\csc x \cot x" />

## Applying the Rules to Trigonometric Functions

Now let's see how to apply these rules in a few examples.

### Combination of Algebra and Trigonometry

Find the derivative of <InlineMath math="y = 2 \sin x + 5x" />.

**Solution:**

We can differentiate this function term by term using the sum rule.

<div className="flex flex-col gap-4">
    <BlockMath math="y' = \frac{d}{dx}(2 \sin x) + \frac{d}{dx}(5x)" />
    <BlockMath math="y' = 2(\cos x) + 5(1)" />
    <BlockMath math="y' = 2 \cos x + 5" />
</div>

### Using the Product Rule

Find the derivative of <InlineMath math="y = 2 \sin x \tan x" />.

**Solution:**

Use the product rule <InlineMath math="y' = u'v + uv'" />.

Let <InlineMath math="u = 2 \sin x" /> and <InlineMath math="v = \tan x" />.

Then <InlineMath math="u' = 2 \cos x" /> and <InlineMath math="v' = \sec^2 x" />.

<div className="flex flex-col gap-4">
    <BlockMath math="y' = (2 \cos x)(\tan x) + (2 \sin x)(\sec^2 x)" />
    <BlockMath math="y' = (2 \cos x)\left(\frac{\sin x}{\cos x}\right) + (2 \sin x)(\sec^2 x)" />
    <BlockMath math="y' = 2 \sin x + 2 \sin x \sec^2 x" />
</div>

### Using the Quotient Rule

Find the derivative of <InlineMath math="y = \frac{1 + \cos x}{\sin x}" />.

**Solution:**

Use the quotient rule <InlineMath math="y' = \frac{u'v - uv'}{v^2}" />.

Let <InlineMath math="u = 1 + \cos x" /> and <InlineMath math="v = \sin x" />.

Then <InlineMath math="u' = -\sin x" /> and <InlineMath math="v' = \cos x" />.

<div className="flex flex-col gap-4">
    <BlockMath math="y' = \frac{(-\sin x)(\sin x) - (1 + \cos x)(\cos x)}{(\sin x)^2}" />
    <BlockMath math="y' = \frac{-\sin^2 x - (\cos x + \cos^2 x)}{\sin^2 x}" />
    <BlockMath math="y' = \frac{-(\sin^2 x + \cos^2 x) - \cos x}{\sin^2 x}" />
    <BlockMath math="y' = \frac{-1 - \cos x}{1 - \cos^2 x} \quad (\text{Pythagorean Identity})" />
    <BlockMath math="y' = \frac{-(1 + \cos x)}{(1 - \cos x)(1 + \cos x)} \quad (\text{Factorization})" />
    <BlockMath math="y' = \frac{-1}{1 - \cos x} = \frac{1}{\cos x - 1}" />
</div>

## Exercises

1. Find the first derivative of <InlineMath math="f(x) = 4x^3 - 5 \cos x" />.
2. Find the first derivative of <InlineMath math="f(x) = \sin x \cos x" />.

### Answer Key

1.  **Solution:**

    Use the subtraction rule to differentiate each term separately.
    
    **Step 1: Differentiate the first term**

    The derivative of <InlineMath math="4x^3" /> using the power rule is <InlineMath math="3 \cdot 4x^{3-1} = 12x^2" />.

    **Step 2: Differentiate the second term**

    The derivative of <InlineMath math="-5 \cos x" /> is <InlineMath math="-5(-\sin x) = 5 \sin x" />.

    **Step 3: Combine the results**

    <BlockMath math="f'(x) = 12x^2 + 5 \sin x" />

    So, the derivative of the function is <InlineMath math="12x^2 + 5 \sin x" />.

2.  **Solution:**

    Use the product rule, <InlineMath math="f'(x) = u'v + uv'" />.
    
    **Step 1: Determine u, v, u', and v'**

    Let <InlineMath math="u = \sin x" /> and <InlineMath math="v = \cos x" />.

    Then, <InlineMath math="u' = \cos x" /> and <InlineMath math="v' = -\sin x" />.

    **Step 2: Apply the Product Rule**

    <div className="flex flex-col gap-4">
        <BlockMath math="f'(x) = (\cos x)(\cos x) + (\sin x)(-\sin x)" />
        <BlockMath math="f'(x) = \cos^2 x - \sin^2 x" />
    </div>

    **Step 3: Use a Trigonometric Identity (Optional)**

    The result can also be simplified using the double angle identity, <InlineMath math="\cos(2x) = \cos^2 x - \sin^2 x" />.

    <BlockMath math="f'(x) = \cos(2x)" />
    
    So, the derivative of the function is <InlineMath math="\cos(2x)" />.