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Differentiate the function.

$ G(y) = \ln \frac {(2y + 1)^5}{\sqrt {y^2 + 1}} $

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00:36

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 6

Derivatives of Logarithmic Functions

Derivatives

Differentiation

Harvey Mudd College

University of Nottingham

Boston College

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.

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In this problem, we are asked to find a derivative of function, g of or using properties of a recti functions. Since we are dividing 2 functions, we can write this 1 as natural log of 2 y plus 1, the fifth minus natural log of square root of y squared plus 1 point now. This is this further could be written as we can take this power 5 and then write it here so pi times, natural log of 2 y plus 1 minus square root is power to the 1 half. So we can write this 1 as 1, half natural log of 1 square plus 4 point now. Let'S take derivative of this simplified form, so g prime of y, isn't equal to at just a constant 5 times 1 over 2 y plus 1 times the r to t inner function, which is 2 minus 1 half of 1 over y square plus 1 multiplied by A derivative of ennfunction, which is 2 y, so we can then write. The answer is 10 divided by 2 y plus 1 minus 2 will cancel out y divided by y squared plus 1.

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