Math Problem Statement

Find the derivatives of the given logarithmic functions using the natural logarithm differentiation rules.

Solution

The image contains exercises related to derivatives of natural logarithmic functions. Let’s work through a few examples step-by-step while using the properties of logarithms to simplify first.

Example Solutions:

41. f(x)=ln(3x)f(x) = \ln(3x)

Using the property of logarithms: ln(ab)=lna+lnb\ln(ab) = \ln a + \ln b, f(x)=ln(3x)=ln(3)+ln(x).f(x) = \ln(3x) = \ln(3) + \ln(x). The derivative is: f(x)=0+1x=1x.f'(x) = 0 + \frac{1}{x} = \frac{1}{x}.


42. f(x)=ln(x1)f(x) = \ln(x - 1)

Using the derivative formula ddx[lnu]=uu\frac{d}{dx}[\ln u] = \frac{u'}{u}, where u=x1u = x - 1 and u=1u' = 1, f(x)=1x1.f'(x) = \frac{1}{x - 1}.


43. g(x)=ln(x2)g(x) = \ln(x^2)

Using the property ln(an)=nln(a)\ln(a^n) = n\ln(a), g(x)=ln(x2)=2ln(x).g(x) = \ln(x^2) = 2\ln(x). The derivative is: g(x)=21x=2x.g'(x) = 2 \cdot \frac{1}{x} = \frac{2}{x}.


44. h(x)=ln(2x2+1)h(x) = \ln(2x^2 + 1)

Using the derivative formula ddx[lnu]=uu\frac{d}{dx}[\ln u] = \frac{u'}{u}, where u=2x2+1u = 2x^2 + 1 and u=4xu' = 4x, h(x)=4x2x2+1.h'(x) = \frac{4x}{2x^2 + 1}.


45. y=(lnx)4y = (\ln x)^4

Using the chain rule, let u=ln(x)u = \ln(x) so y=u4y = u^4. Then, dydx=4u3dudx,\frac{dy}{dx} = 4u^3 \cdot \frac{du}{dx}, where dudx=1x\frac{du}{dx} = \frac{1}{x}. Substituting: dydx=4(lnx)31x=4(lnx)3x.\frac{dy}{dx} = 4(\ln x)^3 \cdot \frac{1}{x} = \frac{4(\ln x)^3}{x}.

Would you like me to proceed with solving more examples or explain any of the steps in further detail?


Questions to Explore:

  1. What are the general rules for logarithmic differentiation?
  2. How can the chain rule and logarithmic properties simplify differentiation problems?
  3. What is the role of implicit differentiation in logarithmic derivatives?
  4. How do we handle derivatives of logarithms involving nested functions?
  5. Can logarithmic differentiation simplify differentiation of complex rational functions?

Tip:

Always simplify logarithmic expressions using log properties before differentiating to make calculations easier!

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Math Problem Analysis

Mathematical Concepts

Differentiation
Natural Logarithms
Logarithmic Properties
Chain Rule

Formulas

d/dx [ln(u)] = u'/u
ln(ab) = ln(a) + ln(b)
ln(a^n) = nln(a)
ln(a/b) = ln(a) - ln(b)

Theorems

Rules of Differentiation
Logarithmic Differentiation

Suitable Grade Level

Grades 11-12 (High School Calculus)