Derivative Of F(x) = 4sec(4x) A Step-by-Step Solution

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In the realm of calculus, finding the derivative of a function is a fundamental operation. Derivatives allow us to understand the rate at which a function's output changes with respect to its input. In this article, we will delve into the process of finding the derivative of the function $f(x) = 4 \sec(4x)$. This function involves the secant function, which is a trigonometric function, and the chain rule, a crucial concept in differentiation. Understanding how to differentiate such functions is essential for various applications in physics, engineering, and other fields.

Understanding the Secant Function and Its Derivative

Before we embark on the differentiation process, it's crucial to have a firm grasp of the secant function and its derivative. The secant function, denoted as $\sec(x)$, is defined as the reciprocal of the cosine function, i.e., $\sec(x) = \frac{1}{\cos(x)}$. Its derivative is a standard result in calculus and is given by:

$$\frac{d}{dx} \sec(x) = \sec(x) \tan(x)$$

This formula forms the basis for differentiating more complex functions involving the secant function. It's also important to remember the derivatives of other trigonometric functions, such as sine, cosine, and tangent, as they often appear in conjunction with the secant function.

Applying the Chain Rule

The function we are dealing with, $f(x) = 4 \sec(4x)$, is a composite function. This means it's a function within a function. To differentiate such functions, we employ the chain rule. The chain rule states that the derivative of a composite function is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function. Mathematically, if we have $f(x) = g(h(x))$, then:

$$f'(x) = g'(h(x)) \cdot h'(x)$$

In our case, we can identify the outer function as $g(u) = 4 \sec(u)$ and the inner function as $h(x) = 4x$. Applying the chain rule will involve differentiating both these functions and combining them appropriately.

Step-by-Step Differentiation of $f(x) = 4 \sec(4x)$

Let's now apply the concepts we've discussed to find the derivative of $f(x) = 4 \sec(4x)$. We will proceed step-by-step to ensure clarity.

Step 1: Identify the Outer and Inner Functions

As we mentioned earlier, the outer function is $g(u) = 4 \sec(u)$ and the inner function is $h(x) = 4x$.

Step 2: Differentiate the Outer Function

We need to find the derivative of $g(u) = 4 \sec(u)$ with respect to $u$. Using the constant multiple rule and the derivative of the secant function, we get:

$$g'(u) = 4 \frac{d}{du} \sec(u) = 4 \sec(u) \tan(u)$$

Step 3: Differentiate the Inner Function

Next, we find the derivative of the inner function $h(x) = 4x$ with respect to $x$:

$$h'(x) = \frac{d}{dx} (4x) = 4$$

Step 4: Apply the Chain Rule

Now we can apply the chain rule, which states $f'(x) = g'(h(x)) \cdot h'(x)$. Substituting our results, we get:

$$f'(x) = 4 \sec(4x) \tan(4x) \cdot 4$$

Step 5: Simplify the Result

Finally, we simplify the expression by multiplying the constants:

$$f'(x) = 16 \sec(4x) \tan(4x)$$

Therefore, the derivative of $f(x) = 4 \sec(4x)$ is $f'(x) = 16 \sec(4x) \tan(4x)$.

Common Mistakes and How to Avoid Them

When differentiating functions involving trigonometric functions and the chain rule, several common mistakes can occur. Being aware of these pitfalls can help you avoid them.

Forgetting the Chain Rule

A frequent error is forgetting to apply the chain rule when differentiating composite functions. It's crucial to identify the inner and outer functions and differentiate them accordingly.

Incorrectly Differentiating Trigonometric Functions

Another mistake is misremembering the derivatives of trigonometric functions. Ensure you have memorized the derivatives of sine, cosine, tangent, secant, cosecant, and cotangent.

Algebraic Errors

Simple algebraic errors, such as incorrect multiplication or simplification, can also lead to wrong answers. Always double-check your calculations.

Sign Errors

Sign errors are common, especially when dealing with derivatives of trigonometric functions. Pay close attention to the signs of the derivatives.

To avoid these mistakes, practice differentiating various functions and carefully review your steps. Using online calculators or software to verify your results can also be helpful.

Applications of the Derivative

The derivative we found, $f'(x) = 16 \sec(4x) \tan(4x)$, has several applications in calculus and related fields. Here are a few examples:

Finding Tangent Lines

The derivative gives the slope of the tangent line to the graph of the function at any point. You can use the derivative to find the equation of the tangent line at a specific point.

Determining Intervals of Increase and Decrease

By analyzing the sign of the derivative, you can determine where the function is increasing or decreasing. Positive values of the derivative indicate an increasing function, while negative values indicate a decreasing function.

Finding Local Maxima and Minima

Critical points, where the derivative is zero or undefined, can be used to find local maxima and minima of the function. These are points where the function reaches a local peak or valley.

Optimization Problems

Derivatives are essential in optimization problems, where the goal is to find the maximum or minimum value of a function subject to certain constraints.

Physics and Engineering

In physics and engineering, derivatives are used to model rates of change, such as velocity and acceleration. They are also used in analyzing oscillations, waves, and other dynamic systems.

Conclusion

In this article, we have thoroughly explored the process of finding the derivative of $f(x) = 4 \sec(4x)$. We began by understanding the secant function and its derivative, then applied the chain rule to differentiate the composite function. We discussed common mistakes and how to avoid them and highlighted several applications of the derivative. Mastering these concepts is crucial for success in calculus and related fields. By practicing and understanding the underlying principles, you can confidently tackle more complex differentiation problems.

Key takeaways from this article include:

• The derivative of $\sec(x)$ is $\sec(x) \tan(x)$.
• The chain rule is essential for differentiating composite functions.
• Common mistakes can be avoided by careful practice and attention to detail.
• Derivatives have numerous applications in mathematics, physics, engineering, and other fields.

By mastering these concepts, you'll be well-equipped to tackle a wide range of calculus problems and apply your knowledge to real-world applications.