   Chapter 11.1, Problem 21E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Find d y d x  if  y  = ln ( t 2 + 3 1 − t ) .

To determine

To calculate: The value of dydt for the function y=ln(t2+31t).

Explanation

Given Information:

The provided function is y=ln(t2+31t).

Formula Used:

The quotient rule for any positive numbers M and N, lnMN=lnMlnN.

The power rule for any positive number M is such that, lnMp=plnM, where p is any real number.

Coefficient rule for a constant c is such that, if f(x)=cu(x), where u(x) is a differentiable function of x, then f(x)=cu(x).

Derivative of natural logarithmic functions is such that, if y=lnu, where u is a differentiable function of x then dydx=1ududx.

Power of x rule for a real number n is such that, if f(x)=xn then f(x)=nxn1.

Constant function rule for a constant c is such that, if f(x)=c then f(x)=0.

Calculation:

Consider the function, y=ln(t2+31t)

Use quotient rule for logarithms,

y=ln(t2+3)ln1t=ln(t2+3)ln(1t)1/2

Use power rule for logarithms,

y=ln(t2+3)12ln(1t)

Differentiate with respect to t,

dydt=ddt[ln(t2+3)12ln(1t)]=ddt[ln(t2+3

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