Problem 38
Question
Find the derivative of the function. $$ g(x)=\tan ^{2}\left(x^{2}+x\right) $$
Step-by-Step Solution
Verified Answer
The derivative of \(g(x) = \tan^2(x^2 + x)\) is:
\(g'(x) = 2\tan(x^2 + x) \cdot \sec^2(x^2 + x) \cdot (2x + 1)\).
1Step 1: Identify the outer and inner functions
The outer function is \(f(u) = \tan^2 u\) and the inner function is \(h(x) = x^2 + x\).
2Step 2: Find the derivative of the outer function
To find the derivative, we can rewrite the outer function as \(f(u) = (\tan u)^2\). Now, use the power rule and the Chain Rule:
\(f'(u) = 2(\tan u)^1 \cdot \dfrac{d}{du}(\tan u) = 2\tan u \cdot \sec^2 u\).
3Step 3: Find the derivative of the inner function
Find the derivative of \(h(x) = x^2 + x\):
\(h'(x) = 2x + 1\).
4Step 4: Apply the Chain Rule
Now, use the Chain Rule to find the derivative of the composite function:
\(g'(x) = f'(h(x)) \cdot h'(x) = 2\tan (x^2 + x) \cdot \sec^2 (x^2 + x) \cdot (2x + 1)\).
5Step 5: Simplify the result
The derivative for the function \(g(x) = \tan^2(x^2 + x)\) is:
\(g'(x) = 2\tan(x^2 + x) \cdot \sec^2(x^2 + x) \cdot (2x + 1)\).
Key Concepts
DerivativeChain RuleTrigonometric FunctionsPower Rule
Derivative
In calculus, a derivative represents the rate at which a function changes at any point. It's like the speedometer of a car, indicating how fast the function's graph is climbing or descending. For the function \( g(x) = \tan^2(x^2 + x) \), the derivative \( g'(x) \) helps us understand how this function's output varies as \( x \) changes.
To get a derivative, especially for complex functions, we use rules such as the Chain Rule and the Power Rule. The process involves differentiating both simple and composite functions to understand their behavior. Derivatives are fundamental to calculus and play a key role in many real-world applications, including physics, engineering, and economics.
To get a derivative, especially for complex functions, we use rules such as the Chain Rule and the Power Rule. The process involves differentiating both simple and composite functions to understand their behavior. Derivatives are fundamental to calculus and play a key role in many real-world applications, including physics, engineering, and economics.
Chain Rule
The Chain Rule is a powerful tool in calculus used to find the derivative of composite functions. A composite function is formed when one function nests inside another, such as \( g(x) = \tan^2(x^2 + x) \).
The essence of the Chain Rule is to differentiate the outside function while keeping the inside function untouched initially, then multiply by the derivative of the inside function.
If you have a function \( y = f(u) \) where \( u = h(x) \), then according to the Chain Rule:
The essence of the Chain Rule is to differentiate the outside function while keeping the inside function untouched initially, then multiply by the derivative of the inside function.
If you have a function \( y = f(u) \) where \( u = h(x) \), then according to the Chain Rule:
- Differentiate the outer function: \( f'(u) \)
- Differentiate the inner function: \( h'(x) \)
- Multiply these derivatives: \( f'(h(x)) \cdot h'(x) \)
Trigonometric Functions
Trigonometric functions are expressions of angles commonly used to relate the angles and sides of right triangles. In calculus, functions like sine, cosine, and tangent are crucial, particularly for problems involving periodic phenomena.
In our example \( g(x) = \tan^2(x^2 + x) \), we use the tangent function, denoted as \( \tan \), and its associated derivative \( \sec^2(u) \), where \( \sec(u) = \frac{1}{\cos(u)} \).
Understanding how to derive trigonometric functions is important:
In our example \( g(x) = \tan^2(x^2 + x) \), we use the tangent function, denoted as \( \tan \), and its associated derivative \( \sec^2(u) \), where \( \sec(u) = \frac{1}{\cos(u)} \).
Understanding how to derive trigonometric functions is important:
- For the tangent function, \( \frac{d}{dx}(\tan x) = \sec^2 x \).
Power Rule
The Power Rule simplifies the process of differentiation for any term in the form of \( x^n \), where \( n \) is any real number. According to this rule, the derivative of \( x^n \) is \( nx^{n-1} \). It's a fundamental rule that helps in breaking down more complex derivatives into simpler expressions.
When differentiating the expression \( f(u) = (\tan u)^2 \) in the problem, we apply the power rule: take the exponent \( 2 \), bring it down, and subtract one from the exponent. This is why the expression becomes \( 2(\tan u)^1 \).
The Power Rule is crucial because it quickly reduces the complexity of expressions, allowing one to focus on applying other rules, like the Chain Rule, for functions with nested parts. It's a basic but highly practical tool for tackling derivatives in calculus challenges.
When differentiating the expression \( f(u) = (\tan u)^2 \) in the problem, we apply the power rule: take the exponent \( 2 \), bring it down, and subtract one from the exponent. This is why the expression becomes \( 2(\tan u)^1 \).
The Power Rule is crucial because it quickly reduces the complexity of expressions, allowing one to focus on applying other rules, like the Chain Rule, for functions with nested parts. It's a basic but highly practical tool for tackling derivatives in calculus challenges.
Other exercises in this chapter
Problem 38
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