Question

Find the derivative of the function. \(g(x)=3 \tan 4 x\)

Short answer

So, the derivative of \(g(x)=3 \tan 4 x\) is \(12 \sec^2 (4x)\).

Step-by-step solution

Find the Derivative of the Inside Function

Here, the inside function is \(4x\). The derivative of \(4x\) is 4.

Find the Derivative of the Outside Function

The outside function is \(\tan (u)\) where \(u = 4x\). The derivative of \(\tan (u)\) is \(\sec^2 (u)\). So, replace \(u\) with \(4x\) to get \(\sec^2 (4x)\).

Apply Chain Rule

According to the Chain Rule, the derivative of a composition of functions is the derivative of the outside function times the derivative of the inside function. Therefore, the derivative of \(3 \tan 4x\) is \(3\) times \(\sec^2 (4x)\) times the derivative of the inside function \(4x\) which is 4.

Multiply the Values

Multiplying the three values, \(3\), \(\sec^2 (4x)\), and 4 together gives the derivative of the function: \(3 * \sec^2(4x) * 4\).

Key concepts

Calculus

Calculus is a branch of mathematics that involves the study of rates of change and accumulation. There are two fundamental concepts in calculus: differentiation and integration. Differentiation is the process of finding the derivative, which measures how a function changes as its input changes. Integration, on the other hand, involves finding an integral, which is the accumulation of quantities over a certain range. Calculus is widely used in science, engineering, economics, statistics, and many other fields to solve problems involving change and motion.

For instance, when studying physics, calculus helps us determine the speed and position of moving objects over time. In economics, it can model how changing conditions affect growth and decay rates. The power of calculus lies in its ability to model and control systems that evolve with time and to compute complex quantities that are otherwise difficult to measure.

Differentiation

Differentiation is a major concept in calculus that deals with the analysis of functions to find their rate of change at any given point. It provides a systematic way to calculate the gradient of a function's graph at any point. This gradient is called the derivative. The process involves certain rules and formulas to compute the derivative of various functions, such as polynomials, trigonometric functions, exponential functions, and logarithms.

Understanding differentiation is crucial because it underlies many physical principles and is fundamental to problem-solving in science and engineering disciplines. It helps us understand how small changes in one quantity lead to changes in another, which is often essential for optimizing outcomes in various real-world scenarios.

Derivative of Tan

The derivative of the tangent function, or 'derivative of tan', is an important trigonometric derivative in calculus. If you have a function involving the tangent of an angle, such as \(\tan(x)\), its derivative is \(\sec^2(x)\), which is the square of the secant of the same angle. A secant function is essentially the reciprocal of the cosine function, \(\sec(x) = \frac{1}{\cos(x)}\).

To differentiate a function like \(\tan(u)\) with respect to \(x\), if \(u\) is a function of \(x\) (say \(u=g(x)\)), we use the Chain Rule. In this way, the derivative of \(\tan(u)\) is the product of the derivative of \(\tan(u)\), which is \(\sec^2(u)\), and the derivative of \(u\) with respect to \(x\). The Chain Rule is a powerful differentiation technique that simplifies dealing with composite functions.

Secant Squared Function

The secant squared function, denoted as \(\sec^2(x)\), is the derivative of the tangent function, but it is also an important function in its own right. The secant function, \(\sec(x)\), is the reciprocal of the cosine function. Then \(\sec^2(x)\) represents the secant function raised to the power of two. In trigonometry, this function is significant because it appears in many calculus problems, especially those involving trigonometric identities.

When you square the secant function, you amplify the magnitude of the original secant function without changing its sign. It's crucial in establishing the rate at which the tangent function is changing and plays a key role in solving integrals involving the tangent function due to its relationship with the derivative of tan (\(\tan(x)\)).