Question

Differentiate the following functions. $$ u=\cos ^{3} x $$

Short answer

Answer: The derivative of the function u = cos^3(x) is u'(x) = -3cos^2(x)sin(x).

Step-by-step solution

Identify the Inner and Outer Functions

In this case, the inner function is the cosine function, f(x) = cos(x), and the outer function is the power function, g(x) = x^3.

Apply the Chain Rule

The chain rule states that if we have a composite function h(x) = g(f(x)), then the derivative h'(x) can be found as follows: $$ h'(x) = g'(f(x)) \cdot f'(x) $$ Here, h(x) = u = g(f(x)) = (cos(x))^3.

Calculate Derivatives of Inner and Outer Functions

First, we calculate the derivatives of the inner and outer functions: - For the inner function, f(x) = cos(x), we have: $$ f'(x) = -\sin(x) $$ - For the outer function, g(x) = x^3, we have: $$ g'(x) = 3x^2 $$

Apply the Chain Rule and Substitute the Calculated Derivatives

Using the chain rule and the calculated derivatives, we find the derivative of the composite function as follows: $$ u'(x) = g'(f(x)) \cdot f'(x) = 3(f(x))^2 \cdot f'(x) $$ Substitute f(x) = cos(x) and f'(x) = -sin(x): $$ u'(x) = 3(cos(x))^2 \cdot (-\sin(x)) $$

Simplify the Final Result

Combining the terms, we get the final result: $$ u'(x) = -3\cos^2(x)\sin(x) $$ So, the derivative of the given function u = cos^3(x) is: $$ u'(x) = -3\cos^2(x)\sin(x) $$

Key concepts

Chain Rule

The chain rule is a fundamental tool in calculus for finding the derivative of composite functions. It helps us to differentiate functions within functions, ensuring we can handle complex problems with ease. When you have a function nested inside another, like in this exercise where you're dealing with \(u = \cos^{3} x\), the chain rule allows us to differentiate in steps. The process involves:

• Identifying the "outer" and "inner" functions.
• Determining their respective derivatives.
• Combining these derivatives to find the overall derivative using the chain rule formula \(h'(x) = g'(f(x)) \cdot f'(x)\).

This method systematizes differentiation problems, making them more manageable and less daunting.

Composite Function

A composite function is simply a function formed by combining two functions, where the output of one function becomes the input of another. It's like layering functions to solve more intricate problems. Understanding composite functions is crucial as it forms the foundation for applying the chain rule effectively.For instance, in \(u = \cos^{3} x\), the inner function is the cosine, \(f(x) = \cos(x)\), and the outer function is the cubic function, \(g(x) = x^3\). Together, they create a composite function where the operations are nested. Recognizing the inner and outer parts helps break down the problem step by step, simplifying the complexity of the function for easier differentiation.

Derivative Calculation

Differentiation is all about finding how a function changes, which is captured by calculating its derivative. In this exercise, we calculated the derivative of \(u = \cos^{3} x\) by breaking down the function into inner and outer layers and using their individual derivatives.

Calculating the Derivatives

• For the inner function \(f(x) = \cos(x)\), the derivative is \(f'(x) = -\sin(x)\).
• For the outer function \(g(x) = x^3\), it becomes \(g'(x) = 3x^2\).

Once both derivatives are known, we combine them using the chain rule, which leads us to the final derivative \(u'(x) = -3 \cos^2(x) \sin(x)\). This process quantifies how the function \(u\) changes with respect to \(x\).

Trigonometric Functions

Trigonometric functions like cosine, sine, and tangent are at the core of many calculus problems. They describe relationships in periodic phenomena and angles in circles, making them indispensable in both theoretical and applied mathematics.In this exercise, the cosine function \(\cos(x)\) plays a significant role as it forms the inner part of the composite function. Knowing and applying the derivative of \(\cos(x)\), which is \(-\sin(x)\), is crucial to solve the problem. The properties of trigonometric functions, such as periodicity and symmetry, often add unique characteristics to their derivatives, providing deeper insights into their behavior under transformation.These properties and derivative rules make trigonometric functions a fascinating and crucial area of study in calculus.