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Differentiate the function.

$ H(z) = \ln \sqrt {\frac {a^2 - z^2}{a^2 + z^2}} $

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$\frac{-2 a^{2} z}{a^{4}-z^{4}}$

00:58

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 6

Derivatives of Logarithmic Functions

Derivatives

Differentiation

Missouri State University

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University of Michigan - Ann Arbor

University of Nottingham

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

02:35

$2-22$ Differentiate the f…

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$ y…

0:00

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02:41

$2-26$ Differentiate the f…

02:44

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02:56

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01:54

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02:03

In this problem, we asked to find to natural log of square root of a square minus c square, divided by a square plus c square. First of all, we know that square root, as we can write this 1 as a square, minus z, square, divided by a square plus z square to the power 1 half using properties of weemin functions. We can take this power and writes multiplied. That is 1 half of natural log of a square, minus c square, divided by a square plus c square. Now we have in the argument of this natural log and we have a division of 2 functions again using properties of natural legrath functions. We can write this 1 as 1, half natural log of a square, minus c square, minus natural log of a square plus z square point. So now we're going to just going to take a derivative of this function. H, prime of x, is then 1 half is just constant. We have 1 over a square minus z squad times the overtop, the inner function, which is negative, 2 z, minus 1 over a square plus c square times the ritt inner function, which is 2 z. Now, let's multiply this term with a square plus c square and multiply this power a square minus z square. We have then 1 half of negative 2 z times a square. Minus 2 z, cube minus 2 z, a square plus 2 z, cube divided by a square, minus z, square, multiplied by a square plus z square. Now these will go away. We have h, prime of c as 1 half of negative 4 a squared times, z, divided by a to the 4 minus c to the 4, is will cancel out and we will have 2 denumerator. So we have. The answer is negative: 2 times a square times. Z, divided by a to the 4 minus z, to our.

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