Problem 29 - Single Variable Calculus

Question

Find the derivative of the function. $$ f(x)=\sin ^{3} x $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = \sin^3{x}\) is \(f'(x) = 3\sin^2{x} \cdot \cos{x}\).

Step 1: Differentiate the inner function

In this case, the inner function is \(u(x) = \sin{x}\). The derivative of the sine function is given by the cosine function. Thus, the derivative of \(u(x)\) is: \[ u'(x) = \frac{d}{dx}\sin{x} = \cos{x} \]

Step 2: Differentiate the outer function

The outer function is \(g(u) = u^3\). We apply the power rule, which states that the derivative of \(u^n\) is \(nu^{n-1}\). Therefore, the derivative of the outer function is: \[ g'(u) = \frac{d}{du}u^3 = 3u^2 \]

Step 3: Apply the chain rule

Now, we apply the chain rule. According to this rule, the derivative of a function composed of two functions (in this case, \(g(u(x))\)) is given by the derivative of the outer function times the derivative of the inner function: \[ f'(x) = g'(u(x))\cdot u'(x) = g'(\sin{x})\cdot \cos{x} \]

Step 4: Substitute the derivatives into the chain rule formula

Lastly, we substitute the derivatives of the inner and outer functions found in steps 1 and 2 into the chain rule formula: \[ f'(x) = 3(\sin{x})^2 \cdot \cos{x} \] So the derivative of the function \(f(x) = \sin^3{x}\) is: \[ f'(x) = 3\sin^2{x} \cdot \cos{x} \]

Key Concepts

Chain Rule

The chain rule is a fundamental tool in calculus used for finding the derivative of composite functions. A composite function is essentially a function within another function. To make the process of differentiation manageable, the chain rule tells us to differentiate the outer function, and then multiply it by the derivative of the inner function. This vital rule simplifies complex calculations.

For example, if you have a function of the form \(f(x) = g(u(x))\), where \(g\) and \(u\) are two functions, the chain rule states that the derivative \(f'(x)\) is obtained by multiplying the derivative of \(g\) with respect to \(u\), denoted as \(g'(u(x))\), by the derivative of \(u\) with respect to \(x\), denoted as \(u'(x)\).

Hence, \(f'(x) = g'(u(x)) \cdot u'(x)\). This simplifies the task of differentiating complex expressions.

Power Rule

The power rule is a basic but crucial differentiation rule in calculus that helps us find the derivative of functions of the form \(x^n\). This applies particularly well to polynomial functions. According to the power rule, the derivative of \(x^n\) is \(nx^{n-1}\). Let's break this down with an example.

Suppose you have \(g(u) = u^3\) as the outer function, as in our exercise. To find its derivative using the power rule, you multiply the exponent \(3\) by \(u\) raised to one power less than before: \(3u^{3-1} = 3u^2\).

By consistently applying this rule, you can differentiate any power of \(x\) quickly and effectively. It forms the basis for differentiating more complex expressions when combined with other rules such as the chain rule.

Trigonometric Functions

Trigonometric functions are key functions in calculus, especially when dealing with periodic or wave-like behaviors. The most common are sine, cosine, and tangent, each having specific derivatives that are commonly used.

The sine function, \(\sin(x)\), which appears in our exercise, has a derivative of \(\cos(x)\). This means every time we differentiate \(\sin(x)\), we get \(\cos(x)\). Similarly, the derivative of \(\cos(x)\) is \(-\sin(x)\). Understanding these derivatives is essential for solving problems involving trigonometric functions.

These relationships are crucial in physics, engineering, and other fields where waveforms are analyzed. Mastering the derivatives of trig functions boosts your problem-solving capabilities.

Calculus Problem Solving

Calculus problems often involve finding derivatives, especially when functions are complex and involve multiple operations. Problem-solving in calculus typically follows a structured approach:

• Identify the type of problem and recognize all the functions involved.
• Determine which rules of differentiation apply, such as the power rule or chain rule.
• Break down the task into manageable steps: differentiate the inner function first, then the outer function, and apply the chain rule if the problem involves composite functions.
• Combine the derivatives systematically and simplify the result if necessary.

In our exercise, we had a function \(f(x) = \sin^3(x)\). Recognizing this as a composition of functions, we applied the chain rule to break it down. This methodical way of tackling calculus problems ensures that complex functions can be managed with ease.