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CalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

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Chapter 4.5, Problem 49E
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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Finding an Equation of a Tangent Line In Exercises 45-52, find an equation of the tangent line to the graph of the function at the given point. See Example 5.

f ( x ) = ln 5 ( x + 2 ) x ; ( − 5 2 , 0 )

To determine

To calculate: The equation of the tangent to the graph of the function f(x)=ln5(x+2)x at the point (−52,0).

Explanation

Given information:

The function is f(x)=ln5(x+2)x and the point is (−52,0).

Formula used:

Let u be a differentiable function of x then, ddx[lnx]=1x,x>0 and ddx(lnu)=1ududx,u>0.

Properties of logarithm,

lnab=lna−lnbln(ab)=lna+lnb

Where, a and b are any real number or variable.

The derivative of function f(x)=un using the chain rule is,

ddx(un)=nun−1dudx

Where, u is the function of x.

Equation of line passes through the point (x1,y1) is given as,

y−y1=m(x−x1)

Where, m is the slope and m=dydx at (x1,y1).

Calculation:

Consider the function, f(x)=ln5(x+2)x.

Apply the properties of logarithmic to the provided function,

ln5(x+2)x=ln5+ln(x+2)−lnx

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Sect-4.1 P-1CPSect-4.1 P-2CPSect-4.1 P-3CPSect-4.1 P-4CPSect-4.1 P-5CPSect-4.1 P-1SWUSect-4.1 P-2SWUSect-4.1 P-3SWUSect-4.1 P-4SWUSect-4.1 P-5SWU
Sect-4.1 P-6SWUSect-4.1 P-7SWUSect-4.1 P-8SWUSect-4.1 P-9SWUSect-4.1 P-10SWUSect-4.1 P-11SWUSect-4.1 P-12SWUSect-4.1 P-13SWUSect-4.1 P-14SWUSect-4.1 P-15SWUSect-4.1 P-16SWUSect-4.1 P-17SWUSect-4.1 P-18SWUSect-4.1 P-1ESect-4.1 P-2ESect-4.1 P-3ESect-4.1 P-4ESect-4.1 P-5ESect-4.1 P-6ESect-4.1 P-7ESect-4.1 P-8ESect-4.1 P-9ESect-4.1 P-10ESect-4.1 P-11ESect-4.1 P-12ESect-4.1 P-13ESect-4.1 P-14ESect-4.1 P-15ESect-4.1 P-16ESect-4.1 P-17ESect-4.1 P-18ESect-4.1 P-19ESect-4.1 P-20ESect-4.1 P-21ESect-4.1 P-22ESect-4.1 P-23ESect-4.1 P-24ESect-4.1 P-25ESect-4.1 P-26ESect-4.1 P-27ESect-4.2 P-1CPSect-4.2 P-2CPSect-4.2 P-3CPSect-4.2 P-4CPSect-4.2 P-5CPSect-4.2 P-1SWUSect-4.2 P-2SWUSect-4.2 P-3SWUSect-4.2 P-4SWUSect-4.2 P-5SWUSect-4.2 P-6SWUSect-4.2 P-7SWUSect-4.2 P-8SWUSect-4.2 P-9SWUSect-4.2 P-10SWUSect-4.2 P-11SWUSect-4.2 P-12SWUSect-4.2 P-1ESect-4.2 P-2ESect-4.2 P-3ESect-4.2 P-4ESect-4.2 P-5ESect-4.2 P-6ESect-4.2 P-7ESect-4.2 P-8ESect-4.2 P-9ESect-4.2 P-10ESect-4.2 P-11ESect-4.2 P-12ESect-4.2 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