Question

Find the second derivative of the function. $$ f(x)=3 \sin x $$

Short answer

The second derivative of the function \(f(x) = 3 \sin x\) is \(f''(x) = -3 \sin x.\)

Step-by-step solution

Identify the Function

First, identify the function that needs to be differentiated, which is \(f(x) = 3 \sin x\). This function is a simple sine function multiplied by a constant value.

Find the First Derivative

Next, calculate the derivative of \(3 \sin x\). The derivative of the sine function is the cosine function, so the first derivative of \(3 \sin x\), denoted as \(f'(x)\) or \(\frac{df}{dx}\), is \(f'(x) = 3 \cos x.\)

Find the Second Derivative

Finally, the second derivative is the derivative of the first derivative. The derivative of a cosine function is negative sine, so the second derivative, denoted as \(f''(x)\) or \(\frac{d^2f}{dx^2}\), will be \(f''(x) = -3 \sin x.\)

Key concepts

Calculus

Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. It is split into two major areas: differential calculus, which concerns the concept of a derivative, and integral calculus, which deals with the concept of an integral. Derivatives describe how a function changes as its input changes, offering a precise formulation for concepts like velocity and slope.

Understanding calculus is crucial for solving complex problems in physics, engineering, economics, and more, as it allows us to model and predict the dynamic changes within these systems. The process of finding a derivative, known as differentiation, is one of the core operations in calculus and is pivotal when discussing motion, growth, and decay among various phenomena.

Differentiation

Differentiation is the process of finding the derivative of a function. To differentiate a function is to determine how its output changes in response to changes in its input value. Essentially, the derivative represents the rate of change or the slope of the function at any given point. For basic functions, there are standard rules of differentiation that one can apply, such as the power rule, product rule, quotient rule, and chain rule.

When functions become more complex, involving multiple operations or composite functions, these rules can be combined to find the derivative. The derivative at a specific point can also be interpreted as the slope of the tangent line to the function's graph at that point, giving a geometric understanding of differentiation.

Sine Function

The sine function, denoted as \(\sin x\), is a fundamental trigonometric function. It is periodic and oscillates between -1 and 1, creating a wave-like pattern on a graph. The function describes the ratio of the opposite side to the hypotenuse in a right-angled triangle, but it also has applications far beyond simple triangle geometry.

In calculus, the sine function's behavior as a function of a real number input relates directly to circular motion and wave phenomena. The fact that \(\sin x\) is differentiable is particularly useful when working with problems involving periodic or oscillatory motion, as it enables the analysis of such systems' instantaneous rates of change.

Cosine Function

Similar to the sine function, the cosine function, denoted as \(\cos x\), is another primary trigonometric function. It also oscillates between -1 and 1, and its graph is a wave that is phase-shifted by \(/frac{\pi}{2}\) units compared to the sine function graph. This means the cosine graph reaches its maximum value when the sine graph crosses the x-axis.

The cosine function represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle. In the context of differential calculus, the importance of the cosine function becomes evident as we find out that it is the first derivative of the sine function with respect to the angle, which creates a deep connection between these two functions.