Unveiling the Derivative of tan 4x: A Step-by-Step Guide
Trigonometric functions are fundamental building blocks in calculus, and understanding their derivatives is crucial for solving a wide range of problems in physics, engineering, and other scientific fields. This article focuses on finding the derivative of tan 4x, a seemingly complex task that simplifies significantly when approached methodically. We'll break down the process step-by-step, ensuring a clear understanding for learners of all levels.
1. Understanding the Chain Rule
Before tackling the derivative of tan 4x, let's refresh our understanding of the chain rule. The chain rule is essential when differentiating composite functions – functions within functions. If we have a function y = f(g(x)), its derivative is given by:
dy/dx = f'(g(x)) g'(x)
In simpler terms, we differentiate the "outer" function, leaving the "inner" function untouched, and then multiply by the derivative of the "inner" function.
2. Identifying the Outer and Inner Functions in tan 4x
In the function y = tan 4x, we can identify two distinct functions:
Outer function: f(u) = tan u (where 'u' represents the inner function)
Inner function: g(x) = 4x
This means our function is a composite function: y = f(g(x)) = tan(4x).
3. Applying the Chain Rule to find the Derivative
Now, we apply the chain rule:
1. Differentiate the outer function: The derivative of tan u with respect to u is sec²u. So, f'(u) = sec²u.
2. Differentiate the inner function: The derivative of 4x with respect to x is 4. So, g'(x) = 4.
3. Combine the results: According to the chain rule, the derivative of tan 4x is:
dy/dx = f'(g(x)) g'(x) = sec²(4x) 4 = 4sec²(4x)
Therefore, the derivative of tan 4x is 4sec²(4x).
4. Practical Examples
Let's solidify our understanding with some examples:
Example 1: Find the slope of the tangent line to the curve y = tan 4x at x = π/8.
First, we find the derivative: dy/dx = 4sec²(4x). Then, we substitute x = π/8:
dy/dx |_(x=π/8) = 4sec²(4 π/8) = 4sec²(π/2)
Since sec(π/2) is undefined (it approaches infinity), the slope of the tangent line at x = π/8 is undefined. This indicates a vertical tangent at that point.
Example 2: Find the derivative of h(x) = 3tan(4x² + 1).
This example involves a more complex inner function. Let's break it down:
Using the chain rule: h'(x) = 3sec²(4x² + 1) 8x = 24x sec²(4x² + 1)
5. Key Takeaways
The chain rule is paramount when differentiating composite functions like tan 4x.
Remember the derivative of tan u is sec²u.
The derivative of tan 4x is 4sec²(4x). Always multiply by the derivative of the inner function.
Practice applying the chain rule with various examples to build confidence and proficiency.
Frequently Asked Questions (FAQs)
1. Why is the chain rule necessary here? The chain rule is crucial because we're dealing with a composite function (a function within a function). The '4x' inside the tangent function requires the chain rule to account for its impact on the overall rate of change.
2. What is sec²(4x)? Sec²(4x) is the square of the secant of 4x. The secant function is the reciprocal of the cosine function, so sec(θ) = 1/cos(θ). Therefore, sec²(4x) = 1/cos²(4x).
3. Can I use other trigonometric identities to simplify the derivative? While the derivative 4sec²(4x) is generally the preferred form, you can use trigonometric identities to express it differently, depending on the context of your problem. For instance, you could use the Pythagorean identity (1 + tan²θ = sec²θ) for further manipulation.
4. What if the inner function was more complex? The chain rule remains the same regardless of the complexity of the inner function. Just differentiate the inner function separately and multiply by the derivative of the outer function.
5. How can I check my work? You can use online derivative calculators or graphing tools to verify your answers. Plotting the original function and its calculated derivative can also provide visual confirmation. Understanding the graphical relationships between a function and its derivative is highly beneficial.
Note: Conversion is based on the latest values and formulas.
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