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Solve the given problems.Find the linearization of the function $f(x)=\frac{6 e^{4 x}}{2 x+3}$ for $a=0$.

Calculus 1 / AB

Chapter 27

Differentiation of Transcendental Functions

Section 6

Derivative of the Exponential Function

Derivatives

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Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

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Okay here we get given a function. That function is F of X equals six. Either the four X over two X plus three. We want the linear ization of this function at a equals zero. So for linear ization we need two things. We need this point and a slope. We'll start off with the point the X coordinate. The point is given to us zero. We need to find the like coordinate. So we're going to compute F of zero. Just six ft of the zero of the zero plus three. Should give us 6/3. Just to coordinate is zero comma. Two will hold on to that later. Okay, now we need the slope so we need the derivative for that. So after kind of X. This will be another closer role. We'll set up on the division symbol, the denominator times the derivative of the numerator. There is the numerator will be 24. Either the four X minus the numerator times the derivative of the denominator all over the denominator square. Okay. We could spend time simplifying this but we don't really need to we just need to evaluate this at our A value so we evaluate it at a equals zero. Okay, I'll skip some of the calculations here. So give us three times 24 ft of the zero minus six times E. To the zero times two. All over two times 00 plus three. It's three squared mm. Little bit more simplification here. 24 times three. We'll have 72 minus 12/9. Should give us 60 of the nine which will simplify to 20/3. Okay. And this is our slope. So how do we put the linear ization together? We use point slope form I minus Y. One equals M. It's x minus X. One. We've got to point up the top and a slope that we've just found substitute these in quick little bit of rearranging can get us to the answer. We'll see in the back of the book, which is why you cause 20/3 X plus two. Just move into two over and distributing the 20/3. And that's it.

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