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106 LearnersLast updated on October 8, 2025
Derivative of Transcendental Functions
We explore the derivatives of transcendental functions, such as \( \tan(x) \). The derivative, \( \sec^2(x) \), serves as a tool to measure how the tangent function changes with respect to small changes in \( x \). Derivatives are crucial in calculating changes, like profit or loss, in real-life situations. We will delve into the derivatives of transcendental functions.
What is the Derivative of Transcendental Functions?
Understanding the derivative of transcendental functions, like ( tan(x) ), is essential in calculus. The derivative of ( tan(x) ) is commonly represented as ( frac{d}{dx} (tan x) ) or ( (tan x)' ), and its value is ( sec^2(x) ). This indicates that the tangent function is differentiable within its domain.
Key concepts include:
Tangent Function: ( tan(x) = frac{sin(x)}{cos(x)} ).
Quotient Rule: A rule for differentiating ( tan(x) ) due to its form ( frac{sin(x)}{cos(x)} ).
Secant Function: ( sec(x) = frac{1}{cos(x)} ).
Derivative of Transcendental Functions Formula
The derivative of ( tan(x) ) can be denoted as ( frac{d}{dx} (tan x) ) or ( (tan x)' ).
The formula for differentiating ( tan(x) ) is: [ frac{d}{dx} (tan x) = sec2 x ]
This formula is valid for all ( x ) where ( cos(x) neq 0 ).
Proofs of the Derivative of Transcendental Functions
The derivative of ( tan(x) ) can be derived using various proofs involving trigonometric identities and differentiation rules.
Methods include:
By First Principle: Using the limit of the difference quotient.
Using Chain Rule: Applying chain differentiation techniques.
Using Product Rule: Breaking down products within functions.
By First Principle
To prove using the First Principle, express the derivative as the limit of the difference quotient for ( f(x) = tan x ): [ f'(x) = lim_{h to 0} frac{tan(x + h) - tan x}{h} ] [ = lim_{h to 0} frac{frac{sin(x + h)}{cos(x + h)} - frac{ sin x}{\cos x}}{h} \] \[ = \lim_{h \to 0} \frac{\sin(x + h)\cos x - \cos(x + h)\sin x}{h \cos x \cos(x + h)} \] Using the identity \( \sin A \cos B - \cos A \sin B = \sin(A - B) \), it becomes: \[ = \lim_{h \to 0} \frac{\sin h}{h \cos x \cos(x + h)} \] Using \( \lim_{h \to 0} \frac{\sin h}{h} = 1 \), it simplifies to \( \frac{1}{\cos^2 x} \). Thus: \[ f'(x) = \sec^2 x \]
Using Chain Rule
For the chain rule: \[ \tan x = \frac{\sin x}{\cos x} \] Using \( f(x) = \sin x \) and \( g(x) = \cos x \), apply the quotient rule: \[ \frac{d}{dx} \left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2} \] Substituting values gives: \[ \frac{\cos^2 x + \sin^2 x}{\cos^2 x} = \frac{1}{\cos^2 x} = \sec^2 x \]
Using Product Rule
Express \( \tan x \) as \( \sin x \cdot (\cos x)^{-1} \) and apply the product rule. Given \( u = \sin x \) and \( v = (\cos x)^{-1} \), use: \[ \frac{d}{dx}[u \cdot v] = u' \cdot v + u \cdot v' \] With \( u' = \cos x \) and \( v' = \frac{\sin x}{\cos^2 x} \): \[ \frac{d}{dx} (\tan x) = \cos x \cdot (\cos x)^{-1} + \sin x \cdot \frac{\sin x}{\cos^2 x} \] Simplifying: \[ = \sec^2 x \]
Higher-Order Derivatives of Transcendental Functions
Higher-order derivatives involve differentiating a function multiple times. They can be complex but provide deeper insights. For a function \( f(x) \), the first derivative \( f'(x) \) indicates the rate of change.
The second derivative \( f''(x) \) shows the rate of change of the rate of change, and this pattern continues. For \( \tan(x) \), successive derivatives highlight how changes evolve over time.
Special Cases
At \( x = \frac{\pi}{2} \), the derivative is undefined due to the vertical asymptote of \( \tan(x) \).
At \( x = 0 \), the derivative of \( \tan x \) is \( \sec^2(0) = 1 \).
Common Mistakes and How to Avoid Them in Derivatives of Transcendental Functions
Errors often occur when differentiating \( \tan(x) \). Understanding correct processes can prevent common mistakes:
Mistake 1
Not Simplifying the Equation
Skipping simplification can result in incorrect outcomes.
Students often jump to the result, especially with the product or chain rule. It's crucial to write each step to avoid errors.
Mistake 2
Forgetting the Undefined Points of \( \tan x \)
Students might overlook that \( \tan x \) is undefined at points like \( x = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots \).
Always consider the function's domain to understand its continuity.
Mistake 3
Incorrect Use of Quotient Rule
Misapplying the quotient rule, such as with \( \frac{\tan x}{x} \), can lead to mistakes. Correct application is: \[ \frac{d}{dx} \left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2} \]
For \( \frac{d}{dx} \left(\frac{\tan x}{x}\right) \), apply: \[ = \frac{x \cdot \sec^2 x - \tan x}{x^2} \]
Mistake 4
Not Writing Constants and Coefficients
Students sometimes forget to multiply constants before \( \tan x \).
For example, \( \frac{d}{dx} (5 \tan x) \) should be \( 5 \sec^2 x \).
Mistake 5
Not Applying the Chain Rule
The chain rule is often neglected, particularly for inner functions.
For example, \( \frac{d}{dx}(\tan(2x)) \) requires considering the derivative of the inner function: \[ = 2 \sec^2(2x) \]
Examples Using the Derivative of Transcendental Functions
Problem 1
Calculate the derivative of \( \tan x \cdot \sec^2 x \).
Given \( f(x) = \tan x \cdot \sec^2 x \), apply the product rule: \[ f'(x) = u'v + uv' \] Here, \( u = \tan x \) and \( v = \sec^2 x \). Differentiate each: \[ u' = \sec^2 x, \quad v' = 2 \sec^2 x \tan x \] Substitute: \[ f'(x) = (\sec^2 x) \cdot (\sec^2 x) + (\tan x) \cdot (2 \sec^2 x \tan x) \] Simplify: \[ f'(x) = \sec^4 x + 2 \sec^2 x \tan^2 x \]
Explanation
The derivative is calculated using the product rule, dividing the function into parts, differentiating them, and combining the results.
Problem 2
A roller coaster's height is modeled by \( y = \tan(x) \). If \( x = \frac{\pi}{6} \) radians, find the slope.
Given \( y = \tan(x) \), differentiate: \[ \frac{dy}{dx} = \sec^2(x) \] Substitute \( x = \frac{\pi}{6} \): \[ \sec^2\left(\frac{\pi}{6}\right) = 1 + \tan^2\left(\frac{\pi}{6}\right) \] Since \( \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \): \[ \sec^2\left(\frac{\pi}{6}\right) = \frac{4}{3} \] The slope is \( \frac{4}{3} \).
Explanation
The slope at \( x = \frac{\pi}{6} \) is calculated by substituting into the derivative \( \sec^2(x) \).
Problem 3
Derive the second derivative of \( y = \tan(x) \).
First derivative: \[ \frac{dy}{dx} = \sec^2(x) \] Differentiate again for the second derivative: \[ \frac{d^2y}{dx^2} = \frac{d}{dx}[\sec^2(x)] \] Using the chain rule: \[ = 2 \sec(x) \cdot \sec(x) \tan(x) \] Simplify: \[ = 2 \sec^2(x) \tan(x) \]
Explanation
The second derivative involves differentiating the first derivative using the product and chain rules.
Problem 4
Prove: \( \frac{d}{dx} (\tan^2(x)) = 2 \tan(x) \sec^2(x) \).
Using the chain rule, consider \( y = \tan^2(x) = (\tan(x))^2 \): \[ \frac{dy}{dx} = 2 \tan(x) \cdot \frac{d}{dx}[\tan(x)] \] Since \( \frac{d}{dx}[\tan(x)] = \sec^2(x) \): \[ \frac{dy}{dx} = 2 \tan(x) \cdot \sec^2(x) \] Hence proved.
Explanation
The chain rule is applied to differentiate \( y = \tan^2(x) \), substituting the derivative of \( \tan(x) \).
Problem 5
Solve: \( \frac{d}{dx} \left(\frac{\tan x}{x}\right) \).
Using the quotient rule: \[ \frac{d}{dx} \left(\frac{\tan x}{x}\right) = \frac{x \cdot \frac{d}{dx}(\tan x) - \tan x \cdot \frac{d}{dx}(x)}{x^2} \] Substitute \( \frac{d}{dx}(\tan x) = \sec^2 x \): \[ = \frac{x \sec^2 x - \tan x}{x^2} \]
Explanation
The function is differentiated using the quotient rule, simplifying the expression to get the final derivative.
FAQs on the Derivative of Transcendental Functions
1.Find the derivative of \( \tan x \).
2.Can we use the derivative of \( \tan x \) in real life?
3.Is it possible to take the derivative of \( \tan x \) at \( x = \frac{\pi}{2} \)?
4.What rule is used to differentiate \( \frac{\tan x}{x} \)?
5.Are the derivatives of \( \tan x \) and \( \tan^{-1} x \) the same?
Important Glossaries for the Derivative of Transcendental Functions
- Derivative: Measures how a function changes as its input changes.
- Tangent Function: The ratio of the opposite to the adjacent side in a right triangle, denoted as \( \tan x \).
- Secant Function: Reciprocal of the cosine function, represented as \( \sec x \).
- First Principle: Method of finding derivatives using the limit of the difference quotient.
- Chain Rule: A rule for differentiating compositions of functions.
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Jaskaran Singh Saluja
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Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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