Problem 30 - Single Variable Calculus

Question

Find the derivative of the function. $$ y=\cos \left(x^{3}\right) $$

Step-by-Step Solution

Answer

The derivative of the function \(y = \cos(x^3)\) is \(\frac{dy}{dx} = -3x^2\sin(x^3)\).

Step 1: Identify the outer and inner functions

The function we're given is a composite function with: - Outer function: \(\cos(u)\), where \(u = x^3\) - Inner function: \(u = x^3\)

Step 2: Differentiate the outer function with respect to the inner function

Now, we need to find the derivative of the outer function, \(\cos(u)\), with respect to \(u\). The derivative of the cosine function is: \(\frac{d}{du}[\cos(u)] = -\sin(u)\)

Step 3: Differentiate the inner function with respect to x

Next, we need to find the derivative of the inner function, \(u = x^3\), with respect to \(x\). The derivative of the cubic function is: \(\frac{du}{dx} = \frac{d}{dx}[x^3] = 3x^2\)

Step 4: Apply the chain rule to find the derivative of the composite function

Now, we'll apply the chain rule to find the derivative of the composite function \(y = \cos(x^3)\) with respect to \(x\). The chain rule states that \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\). Plugging in the derivatives we found in steps 2 and 3, we get: \(\frac{dy}{dx} = -\sin(u) \cdot 3x^2 = -\sin(x^3) \cdot 3x^2\)

Step 5: Simplify the derivative

Finally, we simplify the derivative we found in step 4: \(\frac{dy}{dx} = -3x^2\sin(x^3)\) The derivative of the function \(y = \cos(x^3)\) is \(\frac{dy}{dx} = -3x^2\sin(x^3)\).

Key Concepts

Composite Function

A composite function is created when one function is applied to another. It’s like putting a function inside another function. Imagine a wrapper around a candy. Here, the candy is the inner function, and the wrapper is the outer function. To find the derivative of a composite function, we need to understand both the outer and inner functions involved. For example, in the function \( y = \cos(x^3) \), \( x^3 \) is the inner function wrapped by the cosine function, which is the outer function. Understanding this layering is crucial for applying the chain rule effectively.

Cosine Function

The cosine function is a fundamental trigonometric function vital for studying angles and wave patterns. The cosine of an angle \( x \) gives the ratio of the adjacent side to the hypotenuse in a right triangle. When differentiating the cosine function, we must remember that its derivative results in the negative sine function. Mathematically, this is represented as \( \frac{d}{dx} [\cos(x)] = -\sin(x) \). This transformation is essential when finding derivatives involving trigonometric functions, especially when they form part of composite functions.

Derivative Calculation

Calculating the derivative of a function involves determining how it changes with respect to its variable. This process requires understanding the rate of change or the slope of the function's graph. For composite functions, the chain rule is a key technique. The chain rule states: if \( y = f(g(x)) \), then \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \). This means the derivative of the outer function evaluated at the inner function is multiplied by the derivative of the inner function. Practically, this enables us to handle derivatives of complex, nested functions efficiently.

Cubic Function

A cubic function is a polynomial of degree three, typically written in the form \( ax^3 + bx^2 + cx + d \). These functions have characteristic curves with possible inflection points and varying rates of change. When differentiating a basic cubic function like \( x^3 \), the power rule is applied. This rule helps us find that the derivative of \( x^3 \) is \( 3x^2 \). Understanding this straightforward derivative is a stepping stone to tackling more intricate combinations of functions, like the composite function \( y = \cos(x^3) \). By recognizing the derivatives of simple polynomials, you can build the confidence to work with more complex mathematical expressions.