Question

if \(y=\cos x^{2}+5[(3 / x)+4]^{6}\) then \(d y / d x\) is (a) \(-2 \mathrm{x} \sin \mathrm{x}^{2}+90 \mathrm{x}^{-2}\left(3 \mathrm{x}^{-1}+4\right)^{5}\) (b) \(-2 \mathrm{x} \sin \mathrm{x}^{2}+30 \mathrm{x}^{-2}\left(3 \mathrm{x}^{-1}+4\right)^{5}\) (c) \(+2 \mathrm{x} \sin \mathrm{x}^{2}-90 \mathrm{x}^{-2}\left(3 \mathrm{x}^{-1}+4\right)^{6}\) (d) \(-2 \mathrm{x} \sin \mathrm{x}^{2}-90 \mathrm{x}^{-2}\left(3 \mathrm{x}^{-1}+4\right)^{5}\)

Short answer

The short answer is: (d) \(-2x\sin(x^2) - 90x^{-2}[(3x^{-1})+4]^5\).

Step-by-step solution

Differentiating the first term: \(\cos(x^2)\)

To differentiate the first term, we'll use the chain rule. The chain rule states that if you have a composite function such as \(u(v(x))\), then its derivative with respect to \(x\) is: \(\frac{du}{dx}=\frac{du}{dv}\cdot\frac{dv}{dx}\). Here, let \(u(x) = \cos(v(x))\) and \(v(x) = x^2\). Then, differentiating \(u\) with respect to \(v\), we get: \[\frac{du}{dv} = -\sin(v).\] Now, differentiating \(v\) with respect to \(x\), we get: \[\frac{dv}{dx} = 2x.\]Multiplying these two derivatives, we obtain the derivative of the first term of the function, \(\frac{du}{dx}\): \[\frac{du}{dx}=-\sin(v) \cdot 2x = -2x\sin(x^2).\]

Differentiating the second term: \(5[(3/x)+4]^6\)

To differentiate the second term, we will use both the chain rule and the power rule. First, let's define: \(u(x) = 5 \cdot v(x)^6\) and \(v(x) = (3/x) + 4 \). Now, differentiating \(u\) with respect to \(v\), we get: \[\frac{du}{dv} = 30 v^5.\] Next, differentiate \(v\) with respect to \(x\). Keep in mind that the derivative of \(3/x\) is \(-3/x^2\). We get: \[\frac{dv}{dx} = -\frac{3}{x^{2}}.\] Multiplying these two derivatives, we obtain the derivative of the second term of the function, \(\frac{du}{dx}\): \[\frac{du}{dx} = 30 v^5 \cdot -\frac{3}{x^{2}} = -90 \cdot x^{-2} \cdot [(3/x) + 4]^5.\]

Combining the derivatives

Now that we have the derivatives of the two terms of the given function, we can add them together to find the derivative of the entire function. \[\frac{dy}{dx} = -2x\sin(x^2) - 90 \cdot x^{-2} \cdot [(3/x) + 4]^5.\] Comparing this result with the given options, we find that it matches option (d). Thus, the correct answer is: (d) $-2 \mathrm{x} \sin \mathrm{x}^{2}-90 \mathrm{x}^{-2}\left(3 \mathrm{x}^{-1}+4\right)^{5}$.

Key concepts

Chain Rule

The chain rule is a technique used in calculus to differentiate composite functions. When you have a function nested within another function, like \( y = \cos(x^2) \), the chain rule helps determine the derivative. To apply the chain rule:

• Identify the outer function and the inner function. For \( \cos(x^2) \), the outer is \( \cos(v) \), and the inner is \( v = x^2 \).
• Differentiating the outer function with respect to the inner, you get \( \frac{du}{dv} = -\sin(v) \).
• Differentiating the inner function with respect to \( x \), you obtain \( \frac{dv}{dx} = 2x \).

Multiply these derivatives to get \( \frac{d}{dx} \cos(x^2) = -2x\sin(x^2) \). The chain rule provides a systematic approach to handle such compositions.

Power Rule

The power rule is a straightforward method for differentiating functions of the form \( x^n \). It states that the derivative of \( x^n \) is \( nx^{n-1} \). This rule simplifies differentiation significantly when combined with the chain rule.
For example, in the term \( [ (3/x) + 4 ]^6 \), we:

• Differentiate using the power rule: \( 6[ v(x) ]^5 \) if \( v(x) = (3/x) + 4 \).
• Use the derivative \( \frac{du}{dv} = 30v^5 \) as part of the process.

This rule plays a crucial role when differentiating expressions that are raised to a power.

Trigonometric Differentiation

Trigonometric functions like sine and cosine have specific derivatives that you'll find helpful. For instance:

• The derivative of \( \sin(x) \) is \( \cos(x) \).
• The derivative of \( \cos(x) \) is \( -\sin(x) \).

In our problem, we encounter \( \cos(x^2) \), which requires both trigonometric differentiation and the chain rule. By associating their derivatives with angles themselves, we can easily solve complex trig-based differentiations.
This blend of using trigonometric rules with chain rule principles simplifies derivative calculation for functions involving trigonometrics.

Composite Function

A composite function is formed when one function is applied to the result of another function. In our case, \( y = \cos(x^2) + 5[(3/x)+4]^6 \) is a combination of trigonometric and polynomial forms.
When differentiating composite functions:

• Identify each component function and how they're nested.
• Apply the chain rule to navigate through the layers of functions.

Understanding how composite functions build from simpler ones allows us to differentiate them effectively. This helps to tackle derivatives involving intertwined functions with ease.

Inverse Functions

Inverse functions reverse the effect of a function. While they don’t appear directly in our problem, understanding them is key. For differentiating inverse functions:

• The derivative of an inverse function \( f^{-1}(x) \) is given by \( \frac{1}{f'(f^{-1}(x))} \).
• These derivatives help solve problems involving swaps of dependent and independent variables.

Although not needed here, knowing inverse differentiation provides a full toolkit for handling complex calculus situations.