Question

Finding a Derivative In Exercises \(43-64\) , find the derivative of the function. $$ g(x)=5 \tan 3 x $$

Short answer

The derivative of the function \(g(x)=5 \tan 3x\) is \(g'(x) = 15 \sec^2 3x\).

Step-by-step solution

Identify the function to be differentiated

The function given is \(g(x) = 5 \tan 3x\). Our task is to find its derivative.

Apply chain rule

The chain rule states that the derivative of a composition of functions is the derivative of the outside function evaluated at the inside function, times the derivative of the inside function. In this case, the outside function is \(5 \tan x\) and the inside function is \(3x\). The derivative of \(\tan x\) is \(\sec^2 x\), and the derivative of \(3x\) is \(3\). Substituting, we obtain \(5 \sec^2(3x) \times 3\).

Simplify the expression

Multiplying the constants together, we get the derivative of the function as \(g'(x) = 15 \sec^2 3x\).

Key concepts

Chain Rule in Calculus

The chain rule is a fundamental tool in calculus that provides a method for finding the derivative of composite functions. In essence, it tells us how to differentiate a function that is made up of one function inside another function, typically referred to as the outer function and the inner function.

To apply the chain rule, we take the derivative of the outer function and then multiply it by the derivative of the inner function. A common way to remember this process is by the formula: \( \frac{d}{dx} (f(g(x))) = f'(g(x)) \cdot g'(x) \).

When you encounter a complex expression like \(5 \tan(3x)\), the \(5 \tan(x)\) is the outer function, and \(3x\) is the inner function. The derivative of \(5 \tan(x)\) with respect to \(x\) would normally be \(5 \sec^2(x)\), but since our argument is not just \(x\) but \(3x\), we multiply by the derivative of the inner function, which is \(3\), leading to \(15 \sec^2(3x)\) as the final derivative.

Tangent Function Differentiation

Differentiating the tangent function is a common operation in calculus. The tangent function, represented as \( \tan(x) \), has a derivative that can be calculated using the quotient rule, since \( \tan(x) \) is the quotient of sine and cosine functions, or by using known derivatives of trigonometric functions.

The derivative of the tangent function is \( \sec^2(x) \). This result is essential when dealing with the differentiation of functions that involve the tangent function as part of a composition, such as \( g(x) = 5 \tan(3x) \).

In the exercise provided, once you realize that the outer function is \( 5 \tan(x) \), and you're dealing with tangent function differentiation, it becomes clear that the derivative of that part will be \( 5 \sec^2(x) \). This insight is crucial for applying the chain rule correctly and finding the derivative of the function accurately.

Secant Squared Function

Understanding the secant squared function, denoted as \( \sec^2(x) \), is vital in calculus, especially when dealing with the derivatives of trigonometric functions. The secant function is the reciprocal of the cosine function, and the square indicates that it is raised to the second power.

The relevance of the secant squared function comes from its relationship with the derivative of the tangent function; as seen in the previous section, the derivative of \( \tan(x) \) is \( \sec^2(x) \). This relationship is particularly useful when functions involve the tangent, like in the exercise \( g(x) = 5 \tan(3x) \), as we directly use \( \sec^2(x) \) when applying the chain rule.

After applying the chain rule, we found that the derivative of the given function \( g(x) \) is \( 15 \sec^2(3x) \), which showcases the secant squared function in action within the context of differentiation.