Problem 7 - A Primer of Real Analysis

Question

If \(f(x)=\tan (x)\) and \(g(x)=\cot (x),\) show that $$ f^{\prime}(x)=\sec ^{2}(x) $$ and $$ g^{\prime}(x)=-\csc ^{2}(x) $$

Step-by-Step Solution

Answer

The derivatives are \( f'(x) = \sec^2(x) \) and \( g'(x) = -\csc^2(x) \).

Step 1: Differentiate f(x) = tan(x)

To find the derivative of the function \( f(x) = \tan(x) \), recall the derivative rule for tangent. The derivative of \( \tan(x) \) with respect to \( x \) is \( \sec^2(x) \). Therefore, \( f'(x) = \sec^2(x) \).

Step 2: Differentiate g(x) = cot(x)

To find the derivative of the function \( g(x) = \cot(x) \), recall the derivative rule for cotangent. The derivative of \( \cot(x) \) with respect to \( x \) is \( -\csc^2(x) \). Therefore, \( g'(x) = -\csc^2(x) \).

Step 3: Verify Rules Used

The derivative formulas used for \( \tan(x) \) and \( \cot(x) \) can be derived from the sine and cosine functions. For \( f(x) = \frac{\sin(x)}{\cos(x)} \), using the quotient rule yields \( \sec^2(x) \). For \( g(x) = \frac{\cos(x)}{\sin(x)} \), using the quotient rule yields \( -\csc^2(x) \). These confirm the formulas used in Steps 1 and 2.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics and geometry, especially when dealing with right-angled triangles and periodic phenomena. They relate the angles of a triangle to the lengths of its sides. The primary trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). Each function provides a unique relationship and behaves in a periodic manner, meaning it repeats its values in regular intervals.

In addition to these primary functions, there are the reciprocal trigonometric functions: cosecant (\( \csc \)), secant (\( \sec \)), and cotangent (\( \cot \)). These are defined as

• \( \csc(x) = \frac{1}{\sin(x)} \)
• \( \sec(x) = \frac{1}{\cos(x)} \)
• \( \cot(x) = \frac{1}{\tan(x)} \)

Each function has specific derivatives which are useful for solving various calculus problems.Understanding these functions is key for exploring more complex mathematical concepts like derivatives and integrals.

Tangent Function

The tangent function, noted as \( \tan(x) \), is a trigonomic function that represents the ratio of the sine of an angle to the cosine of that angle. It is expressed as \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). The function oscillates between negative and positive infinity, with undefined points at odd multiples of \( \frac{\pi}{2} \), where the cosine is zero.

One key aspect of \( \tan(x) \) in calculus is its derivative. The derivative of the tangent function is:\[ f'(x) = \sec^2(x) \]This is derived using the quotient rule for differentiation, which is suitable when a function is defined as a ratio of two other functions. The secant squared function provides valuable information about the slope of the tangent line to the curve at any given point.

The tangent function is widely used in various scientific and engineering fields to model periodic behavior and to solve problems involving angles and slopes.

Cotangent Function

The cotangent function, expressed as \( \cot(x) \), is another important trigonometric function. It is defined as the reciprocal of the tangent function, which can be written as:\[ \cot(x) = \frac{\cos(x)}{\sin(x)} \]The function is not defined at integer multiples of \( \pi \), where the sine is zero.

In calculus, the derivative of \( \cot(x) \) plays a crucial role. The derivative of the cotangent function is given by:\[ g'(x) = -\csc^2(x) \]Like the tangent derivative, this result is obtained using the quotient rule for differentiation. The negative cosecant squared function indicates how the slope of the \( \cot(x) \) function behaves as it decreases.

Cotangent is frequently utilized in mathematical equations and real-world applications such as wave mechanics and signal processing, owing to its periodic and inverse characteristics.