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Chapter 6: Problem 10

If \(y=\frac{1}{x}+\frac{2}{x^{2}}+\frac{3}{x^{3}}\), find \(\frac{d y}{d x}\)

Short Answer

Expert verified
-x^{-2}-4*x^{-3}+9*x^{-4}

Step by step solution

01

Rewriting the function y

Rewrite the terms in the given function in the form \(x^n\) to make them amenable to the power rule. So, \(y=x^{-1}+2*x^{-2}+3*x^{-3}\)
02

Apply the Power Rule

Applying the power rule \(d/dx[x^n]=n*x^{n-1}\), the derivative of each term of y with respect to x is calculated as follows: \n\n - \(d/dx[x^{-1}]= -1*x^{-2}\), \n\n - \(d/dx[2*x^{-2}]= -4*x^{-3}\), \n\n - \(d/dx[3*x^{-3}]= 9*x^{-4}\)
03

Simplify the Resultant Derivative

Combine all the derived terms for the complete derivative of y. Therefore, \(\frac{d y}{d x} = -x^{-2}-4*x^{-3}+9*x^{-4}\)

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Most popular questions from this chapter

Let \(f(x)=\left|\begin{array}{lll}\cos x & \sin x & \cos x \\ \cos 2 x & \sin 2 x & 2 \cos 2 x \\ \cos 3 x & \sin 3 x & 2 \cos 3 x\end{array}\right|\) Then find the value of \(f^{\prime}\left(\frac{\pi}{2}\right)\).

Rolle's theorem is applicable for the function \(f(x)=(x-1)|x|+|x-1|\) in the interval (a) \([0,1]\) (b) \(\left[\frac{1}{4}, \frac{3}{4}\right]\) (c) \(\left[-\frac{1}{2}, \frac{1}{2}\right]\) (d) \(\left[\frac{1}{5}, \frac{6}{7}\right]\)

The number of roots of the equation \(\sin x+2 \sin 2 x\) \(+3 \sin 3 x=0\) is the interval \([0, \pi]\) is (a) 1 (b) 2 (c) 3 (d) More than 3

If \(\log (x+y)=2 x y\), then prove that \(y^{\prime}(0)=1\).

Find \(\frac{d y}{d x}\), if \(y=(\tan x)^{\cot x}+(\cot x)^{\tan x}\)
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