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Chapter 6: Problem 8

If \(y=\log _{x} x+10\), find \(\frac{d y}{d x}\)

Short Answer

Expert verified
\(\frac{d y}{d x} = \frac{1-\ln(x)}{x^2}\

Step by step solution

01

Identify the Power Rule, Chain Rule

Before diving into the differentiation, identify the following differentiation rules which will be used in this problem. The Power Rule says that the derivative of \(x^n\) is \(n*x^{n-1}\). The Chain Rule states that the derivative of a composition of functions is the derivative of the outside function times the derivative of the inside function.
02

Apply Chain Rule to the Function

Start with the function \(f(x) = \log _{x} x\). It can be rewritten as \(f(x) = \frac{\ln(x)}{\ln(x)}\). Differentiating it with respect to \(x\), we use the chain rule. The derivative of \(\ln(x)\) is \(\frac{1}{x}\), so the derivative is \(\frac{1}{x} * 1 - \frac{\ln(x)*1}{x^2}.\)
03

Simplifying the Derivative of the Function

The result from the last step can be simplified further to \(\frac{1-\ln(x)}{x^2}\).
04

Differentiate the Constant Term

The derivative of a constant term is 0, so the derivative of 10 is 0.
05

Combine the Derivatives

Now we combine the derivatives calculated in steps 3 and 4 to get the final result. Therefore, \(\frac{d y}{d x} = \frac{1-\ln(x)}{x^2} + 0\

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Most popular questions from this chapter

There is a polynomial \(P(x)=a x^{3}+b x^{2}+c x+d\) such that \(P(0)=P(1)=-2, P^{\prime}(0)=-1\), then find the value of \(a+b+c+d+10\)

If \(y=\cos ^{-1}\left(\frac{5 t+12 \sqrt{1-t^{2}}}{13}\right)\) and \(x=\cos ^{-1}\left(\frac{1-t^{2}}{1+t^{2}}\right)\), find \(\frac{d y}{d x}\)

If \(y=\sin ^{-1} x\), then prove that (i) \(\left(1-x^{2}\right) y_{2}-x y_{1}=0\) (ii) \(\left(1-x^{2}\right) y_{n+2}-(2 n+1) x y_{n+1}-n^{2} y_{n}=0\)

vIf \(y=e^{2 x}\), find \(\left(\frac{d^{2} y}{d x^{2}}\right)\left(\frac{d^{2} x}{d y^{2}}\right)\).

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