Question

Compute the derivative of the given function. $$h(\theta)=\tan \left(\theta^{2}+4 \theta\right)$$

Short answer

\(h'(\theta) = \sec^2(\theta^2 + 4\theta) \cdot (2\theta + 4)\).

Step-by-step solution

Identify the Outer Function

The given function is a composition function where the outer function is the tangent function, \ h(x) = \tan(x). Here, x is the inner function \(\theta^2 + 4\theta\).

Differentiate the Outer Function

The derivative of \(\tan(x)\) is \(\sec^2(x)\). So, the derivative of \(h(\theta)\) with respect to the inner function \(x=\theta^2 + 4\theta\) is \(\sec^2(\theta^2 + 4\theta)\).

Find the Derivative of the Inner Function

The inner function \(x=\theta^2 + 4\theta\) is a polynomial. Differentiate it to get \(\frac{d}{d\theta} (\theta^2 + 4\theta) = 2\theta + 4\).

Apply the Chain Rule

By the chain rule, the derivative of the composite function is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function. Thus, \(h'(\theta) = \sec^2(\theta^2 + 4\theta) \cdot (2\theta + 4)\).

Write Down the Final Result

The derivative of the function \(h(\theta) = \tan(\theta^2 + 4\theta)\) is \(h'(\theta) = \sec^2(\theta^2 + 4\theta) \cdot (2\theta + 4)\).

Key concepts

Chain Rule

The Chain Rule is a fundamental tool in calculus used to differentiate composite functions. A composite function is one where a function is applied to the result of another function. The Chain Rule helps us find the derivative of such functions by decomposing the problem into steps that involve the outer and inner functions separately. Here’s how it works:

• Consider a function composed as \( f(g(x)) \).
• The Chain Rule states that the derivative \( (f(g(x)))' = f'(g(x)) \times g'(x) \).

So, when using the Chain Rule, you first take the derivative of the outer function and evaluate it at the inner function. Then, multiply by the derivative of the inner function itself.
In the given exercise, we used the Chain Rule to differentiate \( h(\theta) = \tan(\theta^2 + 4\theta) \).
The derivative of \( \tan \) (the outer function) is \( \sec^2 \). We evaluate this at the inner function, getting \( \sec^2(\theta^2 + 4\theta) \).
Then, multiply by the derivative of the inner polynomial \( \theta^2 + 4\theta \). This combination—the Chain Rule—efficiently gives us the solution.
Understanding and applying the Chain Rule is crucial for tackling many types of problems in calculus, especially those involving trigonometric and polynomial functions.

Tangent Function

The tangent function, often denoted as \( \tan(x) \), is a fundamental trigonometric function. It represents the ratio of the opposite side to the adjacent side in a right-angled triangle.
Within calculus, the tangent function arises frequently in various applications. When differentiated, the tangent function yields the secant squared function:

• The derivative of \( \tan(x) \) is \( \sec^2(x) \).
• This result is a key part of the Chain Rule solution for composite functions involving \( \tan \).

In this exercise, our "outer function" was \( \tan(\theta^2 + 4\theta) \). Knowing that its derivative is \( \sec^2(\theta^2 + 4\theta) \) was vital for correctly applying the Chain Rule.
The secant squared function, \( \sec^2(x) \), is crucial because it translates changes in the tangent function through its slope.
Understanding how the tangent and secant squared functions relate underscores many advanced calculus topics.

Polynomial Differentiation

Polynomial Differentiation is a straightforward yet powerful technique in calculus. Polynomials consist of terms like \( ax^n \) where \( a \) is a coefficient and \( n \) is a non-negative integer exponent. Differentiating these terms follows a predictable pattern:

• The derivative of \( ax^n \) is \( n \times ax^{n-1} \).

This rule simplifies the process, allowing us to differentiate complex polynomial expressions term by term. In our specific problem, the inner function was a polynomial, \( \theta^2 + 4\theta \).
Applying the basic differentiation rule, we found that its derivative is \( 2\theta + 4 \).

• The term \( \theta^2 \) differentiated becomes \( 2\theta \).
• The term \( 4\theta \) differentiates to simply \( 4 \).

Differentiating polynomials is often one of the first skills learned in calculus because of its elegance and directness.
It lays the groundwork for the efficient use of methods like the Chain Rule in solving more complex functions, such as trigonometric functions with polynomial inputs.