Problem 52 - Calculus for Biology and Medicine

Question

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\left(2 x^{3}-x\right) \cos \left(1-x^{2}\right) $$

Step-by-Step Solution

Answer

The derivative of \( f(x) = (2x^3 - x) \cos(1-x^2) \) is \( (6x^2 - 1)\cos(1-x^2) + (4x^4 - 2x^2)\sin(1-x^2) \).

Step 1: Identify the Rule Needed

Since the function we are given is a product of two functions, we will use the Product Rule to find the derivative. The Product Rule states that if we have a function in the form \( u(x)v(x) \), then the derivative \( (uv)' = u'v + uv' \). Here, \( u(x) = 2x^3 - x \) and \( v(x) = \cos(1-x^2) \).

Step 2: Differentiate \( u(x) \)

Differentiate \( u(x) = 2x^3 - x \) with respect to \( x \). The derivative is \( u'(x) = 6x^2 - 1 \) since the derivative of \( 2x^3 \) is \( 6x^2 \) and the derivative of \( -x \) is \( -1 \).

Step 3: Differentiate \( v(x) \)

Differentiate \( v(x) = \cos(1-x^2) \) with respect to \( x \). Use the Chain Rule. The outer function is \( \cos \), which has a derivative of \( -\sin \), and the inner function is \( 1-x^2 \), which has a derivative of \( -2x \). Thus, \( v'(x) = -\sin(1-x^2) \cdot (-2x) = 2x \sin(1-x^2) \).

Step 4: Apply the Product Rule

Apply the Product Rule using \( u'(x) \) and \( v'(x) \). The derivative \( (uv)' = u'v + uv' = (6x^2 - 1)\cos(1-x^2) + (2x^3 - x)2x\sin(1-x^2) \).

Step 5: Simplify the Expression

Simplify the derivative expression: \( (6x^2 - 1)\cos(1-x^2) + 2x(2x^3 - x)\sin(1-x^2) = (6x^2 - 1)\cos(1-x^2) + (4x^4 - 2x^2)\sin(1-x^2) \). The expression is now simplified.

Key Concepts

Understanding Derivatives

In calculus, the concept of a derivative is fundamental. It helps us determine how a function changes as its input changes. Imagine you're tracking how fast a car is moving. The derivative plays a similar role in mathematics, representing the rate of change of one quantity in relation to another. Here, we are interested in how a function, such as \( f(x) = (2x^3 - x) \cos(1-x^2) \), changes as \( x \) changes.

• The derivative gives the slope of the tangent at any point on the curve of the function.
• For basic functions, like polynomials, the derivative can be found using simple rules: the power rule, the constant rule, and the sum/difference rules.
• For more complex functions that involve products or compositions, we use specific rules like the Product Rule and Chain Rule.

This exercise involves finding the derivative of the product of two functions, which leads us to utilize the Product Rule and Chain Rule for a successful solution.

Applying the Product Rule

The Product Rule is an essential tool in calculus for differentiating functions that are products of two other functions. If you have a function in the form \( f(x) = u(x) \cdot v(x) \), the Product Rule allows us to find the derivative as \( (uv)' = u'v + uv' \).

• In our example, \( u(x) = 2x^3 - x \) and \( v(x) = \cos(1-x^2) \).
• The rule requires us to first find the derivatives of \( u \) and \( v \) individually. This means differentiating each function on its own before combining the results.
• This involves two main steps: differentials of each component function, and then combining these derivatives as per the rule.

The Product Rule is very straightforward and systematic, making it a reliable method to find derivatives in complex expressions involving multiplication of functions.

Using the Chain Rule

The Chain Rule assists us when differentiating composite functions, where one function is inside another. It states that the derivative of a composite function \( f(g(x)) \) can be expressed as \( f'(g(x)) \cdot g'(x) \). In our scenario, we encounter this with the function \( v(x) = \cos(1-x^2) \).

• The outer function here is \( \cos \), and the inner function is \( 1-x^2 \).
• First, differentiate the outer function (\( \cos \)), which gives \( -\sin \). Then, differentiate the inner part (\( 1-x^2 \)), resulting in \(-2x\).
• Multiply the derivative of the outer function by the derivative of the inner one, forming \(2x \sin(1-x^2)\).

The Chain Rule is a powerful concept that enables us to navigate through layers of functions seamlessly, ensuring we can differentiate complicated compositions successfully. Understanding and practicing this rule are vital for tackling advanced differentiation problems efficiently.