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Chapter 3: Problem 12

Find the derivative. \(f(x)=3 x^{2} \sec x-x^{3} \tan x\)

Short Answer

Expert verified
The derivative is \(f'(x) = 6x \sec x + x^2 \sec x \tan x - x^3 \sec^2 x\).

Step by step solution

01

Identify the Parts for Differentiation

The function is given as sum (or difference) of two terms: \(3x^2 \sec x\) and \(x^3 \tan x\). We need to differentiate each term separately.
02

Differentiate the First Term

To differentiate \(3x^2 \sec x\), use the product rule. Let \(u = 3x^2\) and \(v = \sec x\).First, find the derivatives, \(u' = 6x\) and \(v' = \sec x \tan x\). Using the product rule: \[\frac{d}{dx}(uv) = u'v + uv' = 6x \sec x + 3x^2 \sec x \tan x\]
03

Differentiate the Second Term

For \(x^3 \tan x\), also use the product rule. Let \(u = x^3\) and \(v = \tan x\).Find the derivatives: \(u' = 3x^2\) and \(v' = \sec^2 x\).Using the product rule:\[\frac{d}{dx}(uv) = u'v + uv' = 3x^2 \tan x + x^3 \sec^2 x \]
04

Combine the Derivatives

The original function's derivative is the sum of the derivatives found in the previous steps:\[f'(x) = (6x \sec x + 3x^2 \sec x \tan x) - (3x^2 \tan x + x^3 \sec^2 x) \]Simplify the expression:\[f'(x) = 6x \sec x + 3x^2 \sec x \tan x - 3x^2 \tan x - x^3 \sec^2 x\]Combine like terms when possible.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Product Rule
In calculus, the product rule is a fundamental tool for differentiating products of functions. When you have two differentiable functions, say \( u \) and \( v \), and you want to find the derivative of their product \( uv \), the product rule provides a clear path:\[\frac{d}{dx}(uv) = u'v + uv'\]Here's how it unfolds practically:
  • First, differentiate \( u \) to obtain \( u' \).
  • Then, differentiate \( v \) to get \( v' \).
  • Multiply \( u' \) by \( v \), and \( u \) by \( v' \).
  • Finally, add the two resulting expressions together.
In our original exercise, we applied the product rule separately to each term of the function \( f(x) = 3x^2 \sec x - x^3 \tan x \). This rule is essential when handling products within derivatives, ensuring each part is correctly differentiated.
Trigonometric Differentiation
Trigonometric differentiation comes into play when your function involves trigonometric functions like \( \sec x \) or \( \tan x \). These functions have specific derivative rules you need to memorize:
  • The derivative of \( \sec x \) is \( \sec x \tan x \).
  • The derivative of \( \tan x \) is \( \sec^2 x \).
Using these rules is critical when applying the product rule in our exercise. For example, differentiating the part \( 3x^2 \sec x \) involves recognizing that \( \sec x \) transforms into \( \sec x \tan x \) when differentiated. Similarly, \( \tan x \) becomes \( \sec^2 x \). Knowing these trigonometric derivatives allows accurate and efficient problem-solving, ensuring each function is broken down correctly.
Calculus Problem-Solving
Solving calculus problems typically involves a series of logical and systematic steps. Here's a structured approach that you can apply when faced with a differentiation problem:
  • **Identify Components:** Break down the complex function into simpler parts, as seen with \( 3x^2 \sec x \) and \( x^3 \tan x \).
  • **Apply Rules Appropriately:** Use the product rule, chain rule, or quotient rule as needed, considering the specific function characteristics.
  • **Simplify:** After differentiating, always look to simplify the derivative for a cleaner result. This involves combining like terms, factoring, or reducing expressions.
In the original exercise, combining derivatives of separated terms and simplifying the final expression ensures you communicate the most simplified form of the derivative. This process demonstrates not only your skill but also a thorough understanding of calculus concepts, making problem-solving a more straightforward and rewarding experience.

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