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Chapter 10: Problem 830

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

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

Step by step solution

01

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, \(\frac{du}{dx}\): \[\frac{du}{dx}=-\sin(v) \cdot 2x = -2x\sin(x^2).\]
02

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, \(\frac{du}{dx}\): \[\frac{du}{dx} = 30 v^5 \cdot -\frac{3}{x^{2}} = -90 \cdot x^{-2} \cdot [(3/x) + 4]^5.\]
03

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) $-2 \mathrm{x} \sin \mathrm{x}^{2}-90 \mathrm{x}^{-2}\left(3 \mathrm{x}^{-1}+4\right)^{5}$.

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Most popular questions from this chapter

Derivative of function \(\mathrm{f}(\mathrm{x})\left[\mathrm{x}^{2} /\left(1+\sin ^{2} \mathrm{x}\right)\right]\) is (a) Even function (b) Odd function (c) Not define (d) Increasing Function

\((\mathrm{d} / \mathrm{d} \mathrm{x})\left[\log (1+\sin \mathrm{x})+\log (\sec \\{(\pi / 4)-(\mathrm{x} / 2)\\})^{2}\right]=\) (a) 0 (b) \(4 \mid[(\cos x-\tan x) /(\sin x+\cos x)]\) (c) \(\log _{\mathrm{e}} 2\) (d) \(-\log _{\mathrm{e}} 2\)

In \(\mathrm{a}+\mathrm{b}+\mathrm{c}=0\), than the equation \(3 \mathrm{ax}^{2}+2 \mathrm{bx}+\mathrm{c}=0\) has, in the interval \((0,1)\) (a) at least one root (b) at most one root (c) no root (d) Exactly one root exist.

\(\sqrt\left[\left(3 x^{2}+x+1\right) / x\right] \text { then }[d y / d x]_{(x=1)}=\) (a) \(\\{1 / \sqrt{5}\\}\) (b) \(\sqrt{5}\) (c) 5 (d) \((1 / 5)\)

If \(\mathrm{f}(\mathrm{x})=\mathrm{Ax}+\mathrm{B}\) and \(\mathrm{f}(0)=\mathrm{f}^{\prime}(\mathrm{x})=2\) then \(\mathrm{f}(1)=\) (a) 4 (b) 2 (c) 1 (d) \(-4\)
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