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117 LearnersLast updated on September 12, 2025
Derivative of Derivative: Tan x
The derivative of tan(x), which is sec²(x), helps us understand how the tangent function changes. But what about the derivative of this derivative? Exploring these higher-order derivatives gives us more insights into the nature of functions. We will delve into the derivative of the derivative of tan(x) in detail.
What is the Derivative of the Derivative of Tan x?
The first derivative of tan x is sec²x. Differentiating sec²x gives us the second derivative of tan x. This is represented as d²/dx²(tan x) or (tan x)''.
By differentiating sec²x, we find: -
First derivative: sec²x
Second derivative: 2 sec²xtan x
Key concepts:
Tangent Function: tan(x) = sin(x)/cos(x)
Secant Function: sec(x) = 1/cos(x)
Differentiation Rules: Product and Chain Rule
Derivative of the Derivative of Tan x Formula
The first derivative of tan x is sec²x.
To find the derivative of sec²x, use: d²/dx²(tan x) = d/dx(sec²x) = 2 sec²x tan x.
This formula applies to all x where cos(x) ≠ 0.
Proofs of the Derivative of the Derivative of Tan x
To prove the second derivative of tan x, we differentiate sec²x. Methods include: Using Product Rule: -
First derivative: sec²x = sec x · sec x
Use product rule: d/dx(u·v) = u'v + uv'
Differentiate: d/dx(sec x) = sec x tan x - Proof: 2 sec²x tan x
Using Chain Rule:
Differentiate sec²x as (sec x)²
Use chain rule: dy/dx = 2 sec x · d/dx(sec x)
Result: 2 sec²x tan x
Higher-Order Derivatives of Tan x
Higher-order derivatives extend beyond the first and second derivatives. Each subsequent derivative provides deeper insights, similar to analyzing a car's changing speed and acceleration.
For the nth derivative of tan x, we use fⁿ(x). The second derivative, f''(x), shows the rate of change of the rate of change.
Special Cases:
- At x = π/2, derivatives are undefined due to vertical asymptotes. - At x = 0, the first derivative of tan x is sec²(0), which is 1.
Common Mistakes and How to Avoid Them in Derivatives of Derivatives
Mistakes in finding higher-order derivatives often arise from misunderstandings. Here's how to avoid them:
Mistake 1
Not Simplifying Correctly
Ensure each step is simplified correctly. Skipping steps, especially when applying rules, can lead to errors. Always write out each calculation step for clarity.
Mistake 2
Misapplying Differentiation Rules
Misuse of product or chain rules is common. Make sure to apply the correct rule for each step in the differentiation process.
Mistake 3
Forgetting Undefined Points
Remember that tan x is undefined at π/2, 3π/2, etc. Consider the domain when differentiating, as tan x is not continuous at these points.
Mistake 4
Ignoring Higher-Order Derivative Notations
Keep track of derivative notations, especially when differentiating multiple times. Ensure clarity between first, second, and higher-order derivatives.
Mistake 5
Incorrectly Differentiating Composite Functions
Composite functions require careful application of rules. For example, differentiating tan(2x) involves using the chain rule correctly.
Examples Using the Derivative of the Derivative of Tan x
Problem 1
Calculate the second derivative of (tan x·sec²x).
Given f(x) = tan x · sec²x, find the second derivative using the product rule:
First derivative: f'(x) = sec²x · sec²x + tan x · 2 sec²x tan x
Second derivative: Differentiate each term again
Result: f''(x) = 4 sec²x tan²x + 2 sec⁴x
Explanation
The process involves finding the first derivative using the product rule, then differentiating again to find the second derivative.
Problem 2
A slope is modeled by y = tan(x). If x = π/4, find the second derivative of the slope function.
First derivative: dy/dx = sec²(x) Second derivative:
d²y/dx² = 2 sec²(x) tan(x) At x = π/4:
sec²(π/4) = 2 - tan(π/4) = 1
Result: 2 * 2 * 1 = 4
Explanation
We calculate the second derivative by differentiating sec²x and substituting x = π/4 to find the specific value.
Problem 3
Derive the second derivative of y = tan(x) using the chain rule.
First derivative: dy/dx = sec²(x) Second derivative: d²y/dx² = d/dx(sec²(x))
Use chain rule: d²y/dx² = 2 sec(x) · sec(x) tan(x)
Result: 2 sec²(x) tan(x)
Explanation
The chain rule is applied to differentiate sec²x, resulting in the second derivative 2 sec²(x) tan(x).
Problem 4
Prove: d²/dx²(tan²(x)) = 4 tan(x) sec²(x) + 2 sec⁴(x).
Start with y = tan²(x)
First derivative: dy/dx = 2 tan(x) sec²(x)
Second derivative: Apply product rule to dy/dx
Result: 4 tan(x) sec²(x) + 2 sec⁴(x)
Explanation
Using product and chain rules, the proof involves differentiating tan²(x) twice, resulting in the second derivative.
Problem 5
Solve: d²/dx²(tan x/x).
First, find the first derivative using the quotient rule:
d/dx(tan x/x) = (x sec²x - tan x) / x² Differentiate again for the second derivative:
Apply quotient rule to the first derivative.
Result: Differentiating (x sec²x - tan x) / x² involves careful application of the quotient rule.
Explanation
The solution requires applying the quotient rule twice, first for the first derivative, then for the second derivative.
FAQs on the Derivative of the Derivative of Tan x
1.Find the second derivative of tan x.
2.Can the derivative of the derivative of tan x be used in real life?
3.Is it possible to differentiate tan x at x = π/2?
4.What rule is used to find the second derivative of tan x/x?
5.Are the second derivatives of tan x and tan⁻¹x the same?
Important Glossaries for the Derivative of the Derivative of Tan x
- Second Derivative: The derivative of the derivative, showing how the rate of change itself changes.
- Chain Rule: A rule for differentiating composite functions, crucial for finding higher-order derivatives.
- Product Rule: A rule for differentiating products of functions, used for complex derivative calculations.
- Asymptote: A line that a graph approaches but never touches or crosses, affecting continuity.
- Quotient Rule: A rule for differentiating ratios of functions, essential for complex derivatives.
Jaskaran Singh Saluja
About the Author
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.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
