Problem 192 - Calculus Volume 1

Question

Show that \(y=\sec ^{2} x\) and \(y=\tan ^{2} x\) have the same derivative. What can you say about \(y=\sec ^{2} x-\tan ^{2} x ?\)

Step-by-Step Solution

Answer

The derivatives of both functions are the same: \(2 \sec^2 x \tan x\). For \(y=\sec^2 x-\tan^2 x\), the derivative is 0 since it simplifies to 1.

Step 1: Differentiate \(y = \sec^2 x\)

To find the derivative of \(y = \sec^2 x\), we use the chain rule. Recall that \(\sec x\) is \(\frac{d}{dx}(\sec x) = \sec x \cdot \tan x\). So, using the chain rule: \(\frac{d}{dx}(\sec^2 x) = 2 \cdot \sec x \cdot \sec x \cdot \tan x = 2 \sec^2 x \tan x\).

Step 2: Differentiate \(y = \tan^2 x\)

Now, we differentiate \(y = \tan^2 x\) using the chain rule. Recall that the derivative of \(\tan x\) is \(\sec^2 x\). So: \(\frac{d}{dx}(\tan^2 x) = 2 \cdot \tan x \cdot \sec^2 x\).

Step 3: Compare the Derivatives

Observe that both derivatives are actually the same: For \(y = \sec^2 x\), the derivative is \(2 \sec^2 x \tan x\).For \(y = \tan^2 x\), the derivative is \(2 \tan x \sec^2 x\). Since multiplication is commutative, these derivatives are identical.

Step 4: Simplify \(y=\sec^2 x - \tan^2 x\)

Use the trigonometric identity \(\sec^2 x - \tan^2 x = 1\). Therefore, the function \(y = \sec^2 x - \tan^2 x\) simplifies to the constant 1.

Step 5: Derivative of a Constant

The derivative of any constant is 0. Since \(y = \sec^2 x - \tan^2 x\) simplifies to 1, its derivative is \(\frac{d}{dx}(1) = 0\).

Key Concepts

Trigonometric Identities

In trigonometry, identities are equations that hold true for all possible values of the variable. A fundamental example is the Pythagorean identity

• \[ \sec^2 x = \tan^2 x + 1 \].

This identity helps simplify complex expressions involving secant and tangent functions.
In our exercise, we utilized this particular identity to demonstrate that \(y = \sec^2 x - \tan^2 x\) simplifies to a constant value of 1.
By expressing \(\sec^2 x\) as \(\tan^2 x + 1\), the equation \(\sec^2 x - \tan^2 x\) naturally reduces to 1.
Trigonometric identities make manipulation and simplification of trigonometric expressions more manageable and insightful. They are essential tools in calculus when modeling periodic phenomena and simplifying derivatives or integrals involving trigonometric functions.

Chain Rule

The chain rule is a crucial technique in calculus used to differentiate composite functions.
In simple terms, when you have a function nested inside another function, the chain rule allows you to find the derivative of the overall composition.
Consider

• \[ y = (f(g(x))) \]

The derivative, using the chain rule is:

• \[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \].

In our exercise, we used the chain rule to find the derivatives of \(y = \sec^2 x\) and \(y = \tan^2 x\).
For \(y = \sec^2 x\), \(g(x) = \sec x\) and \(f(g) = g^2\). The derivative becomes \(2 \sec x \cdot \sec x \cdot \tan x = 2 \sec^2 x \tan x\).
Similarly, for \(y = \tan^2 x\), \(g(x) = \tan x\) and \(f(g) = g^2\). The derivative is \(2 \tan x \cdot \sec^2 x\).
The chain rule enables differentiating more complex expressions in a structured way.

Constant Function

A constant function is a type of mathematical function in which the output value is the same for any input value.
It is written as \(f(x) = c\) where \(c\) is a constant.
In calculus, the derivative of a constant function is always zero.
This is because the constant function's graph is a horizontal line, meaning there is no change in \(y\) for changes in \(x\).
In the context of our exercise, after simplifying \(y = \sec^2 x - \tan^2 x\) using trigonometric identities, we found that it equals 1, a constant.
Therefore, its derivative is \( \frac{d}{dx}(1) = 0 \).
Understanding constant functions and their derivatives is foundational for solving more complex calculus problems.