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Find the derivatives of $y_{1}=\ln \left(x^{2}\right)$ and $y_{2}=2 \ln x,$ and evaluate these derivatives for $x=-1 .$ Explain your results.

Calculus 1 / AB

Chapter 27

Differentiation of Transcendental Functions

Section 5

Derivative of the Logarithmic Function

Derivatives

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So we're tasked with solving for the derivative of F of X, where f of X is equal to X squared times the quantity of one minus t natural log X squared. So we need to use the product rule here. So we start with or X squared, multiplied by the derivative of the insight of the other function which is one minus natural law. Durex squared. So we're gonna multiply this by the derivative of one since one is a constant will then go to zero in the derivative of the negative natural. Lagerback's negative one times the natural. Lagerback's How you can view it is one over the function times the truth of the inside function. So we start with negative one get negative one. We just rewrite that negative one over X squared times he derivative of X squared, which is to X because of the general. Now we can cancel out the X squared with the two acts and that will simplify too. X squared tries negatives to you have to over X. To simplify this even further, we can cancel out the X squared outside with the axe on the bottom and that will leave us with negative two X Now we just need to So that's the first part. Now we need to add the second function. Ages one my misty natural log of X squared times the derivative But the first, which is X squared and weak, Know that the derivative of any X rays to a power is the power Times X raised to the power minus one. Or in this case, it will just be two x so we can expand this it's to equal. So this is our just basic form of F prime of X equal to negative two x plus the quantity of one times the natural log of X squared times two X But we can simplify this further expand the second half of the equation. So we get negative two X worth. Two acts I s two X, that's your wallet X squared. And so the next the negative to accent positive. Two acts here will cancel out. Move down. It's too little bits. You re right. This to that f prime of axe is equal to negative two x times The natural log of X squared

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