Problem 43 - Calculus for Biology and Medicine

Question

Find the derivative with respect to the independent variable. $$ f(x)=\frac{1}{\tan ^{2} x-\sec ^{2} x} $$

Step-by-Step Solution

Answer

The derivative of \( f(x) \) is 0.

Step 1: Analyse the Function

The function provided is \( f(x) = \frac{1}{\tan^{2}x - \sec^{2}x} \). To differentiate this function, we first need to correctly recognize the components of the equation that involve trigonometric identities. Recall that \( \sec^2 x \) can be expressed as \( 1 + \tan^2 x \).

Step 2: Simplify the Denominator Using Trigonometric Identity

Knowing that \( \sec^2 x = 1 + \tan^2 x \), we substitute this into the denominator: \( \tan^2 x - \sec^2 x = \tan^2 x - (1 + \tan^2 x) = -1 \). So, the function simplifies to \( f(x) = \frac{1}{-1} = -1 \).

Step 3: Differentiate the Simplified Function

Since \( f(x) = -1 \) is a constant function, its derivative with respect to \( x \) is 0. Therefore, \( \frac{df}{dx} = 0 \).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools used in mathematics to simplify expressions involving trigonometric functions such as sine, cosine, and tangent. In our exercise, one key identity used is the relation between the tangent and secant functions: \( \sec^2 x = 1 + \tan^2 x \). This identity is known as the Pythagorean identity for secant and tangent. It reveals a fundamental link between these trigonometric functions and is crucial in expressing or simplifying equations.Understanding identities like this can greatly simplify many mathematical problems, particularly in calculus. For example, by rewriting \( \sec^2 x \) in terms of \( \tan^2 x \), we can often transform a complex expression into something much easier to handle. Such transformations make calculation and further mathematical manipulation far more manageable.

Differentiation Techniques

Differentiation techniques allow us to find the rate at which a function changes concerning its variable, typically denoted as \( x \). In this exercise, once we applied the trigonometric identity, the function \( f(x) = \frac{1}{\tan^2 x - \sec^2 x} \) simplified significantly to \( f(x) = -1 \).For differentiating a constant,

• The derivative of any constant is always 0.
• This is because a constant does not change, so its rate of change over any variable is zero.

Therefore, the derivative of our simplified function with respect to \( x \) is \( 0 \). Recognizing when to apply identities and simplifications is crucial in calculus as it can swiftly lead to simpler differentiation processes.

Simplification of Expressions

Simplification is a technique used to reduce a mathematical expression to its simplest form, making it easier to understand and work with. In calculus, simplification often involves using algebraic manipulation and identities to transform an expression. For example, the original function in this problem, \( f(x) = \frac{1}{\tan^2 x - \sec^2 x} \), appeared complex.By recognizing the Pythagorean trigonometric identity and substituting \( \sec^2 x \) with \( 1 + \tan^2 x \), we simplified the expression:

• Substituted to get \( \tan^2 x - (1 + \tan^2 x) \)
• Which resulted in \( -1 \)

Once simplified, the problem became straightforward. Simplification is especially important in differentiation as it can transform a difficult problem into a much easier one.