Category 3: Logarithmic expansions 1.Function: f(x)=ln?(x?(x+1)) 3. Function: f(

Question

Category 3: Logarithmic expansions

1.Function: f(x)=ln?(x?(x+1)) 3. Function: f(x)=ln?(x^2 tan?(5x))
Expansion: f(x)=ln?(x)+1/2 ln?(x+1) Expansion: f(x)=2 ln?(x)+ln?(tan?(5x))
Derivative: f^' (x)=1/x+1/(2(x+1)) Derivative: f^' (x)=2/x+(5?sec?^2 (x))/(tan?(5x))


2.Function: f(x)=ln(x^2/(x-1)) 4. Function: f(x)=ln ?x/(csc?(x))
Expansion: f(x)=ln?(x^2 )-ln?(x-1) Expansion: f(x)=1/2 ln?(x)-ln?(sec?(x))
Derivative: f^' (x)=2/x-1/(x-1) Derivative: f^' (x)=1/2x-(sec?(x)tan?(x))/(sec?(x))=1/2x-tan?(x)



5.What rules are governing the expansion of the function so that we can use the formula for
the derivative of ln(u) to find the derivative?

Explanation / Answer

Logarithmic differentiation is a tool used to calculate derivatives of products or quotients of functions because of the nice properties of logarithms, especially of products and quotients. Say we want to differentiate some function f(x)g(x)h(x). Instead of using an ugly product rule, let u(x) = f(x)g(x)h(x), then take the natural log of both sides: ln(u(x)) = ln(f(x)g(x)h(x)) = ln(f(x)) + ln(g(x)) + ln(h(x)), which can easily be differentiated using just the chain rule

Related Questions

Navigate

• Browse (All)
• Browse C
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.