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Chapter 5: Problem 49

Finding a Derivative In Exercises \(39-60,\) find the derivative of the function. $$y=\log _{3}\left(x^{2}-3 x\right)$$

Short Answer

Expert verified
\(y^\prime = \frac{2x-3}{(x^{2}-3x) \ln 3}\)

Step by step solution

01

Express the logarithm in terms of natural log

The given function can be written as in terms of natural log as follows: \( y = \ln (x^{2}-3x) / \ln 3 \)
02

Apply the chain rule for derivation

The derivative of this function can be derived using the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function multiplied by the derivative of the inner function. Here, \(u=x^{2}-3x\) and \(g(u)= \ln u / \ln 3\). The derivative of \(y\) is then: \( y' = g'(u)*u' \)
03

Calculate the derivative

The derivative of \( y= \ln u / \ln 3 \) is \( g'(u) = 1 / (u \ln 3) \). The derivative of \( u=x^{2}-3x \) is \( u'= 2x-3 \). Therefore, the derivative of the given function becomes: \( y' = (1 / (u \ln 3)) * (2x-3) = \frac{2x-3}{(x^{2}-3x) \ln 3}\)

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