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Find the derivatives of the given functions.$$u=4 \sqrt{\ln 2 t+e^{2 t}}$$

$\frac{d u}{d t}=\frac{2\left(1+2 t e^{2 t}\right)}{t \sqrt{\ln 2 t+e^{2 t}}}$

Calculus 1 / AB

Chapter 27

Differentiation of Transcendental Functions

Section 6

Derivative of the Exponential Function

Derivatives

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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we want to find the derivative. Dy dx for the function. Why is it with four times the square root? Natural longer than two X. Plus either two X. To do. So we're going to rely on during of shortcuts we picked up from single variable. Help specifically rule 0 to 3 here on the left rule zero is D D X. B. To the U S B D U L N B D X D X E D U E D U D U D X. Rules +123 of the powerful product rule in general respectively. Let's first rewrite. Why? So it's more easily differentiable. Mhm. Why is equal to four times Ellen to expose either two X reached the power of one half. Now we see how the power rule chain rule Come in handy along with Real zero. So remember that the derivative of the natural algorithm of you is do you over you? So we have about the product rule. Do I. D X equals four times one half Ln two X. Let's eat of the two X negative one half times the river. What's inside to over to express to you? The two X times two. Don't forget the term of two on the eve of the two weeks ending because of the chain rule. Thus our final solution is two Plus four x. either two x over X square root of natural algorithm of two X. Let's either the two X or as highlighted now as a final solution.

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