AboutPaul Klarreich Expertise All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions.
I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.
Experience I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.
Expert: Paul Klarreich Date: 6/22/2008 Subject: Basic derivatives
Question I have a couple of questions regarding my calculus homework. I have already
completed most of the assignment, but these 6 problems are what I am
having trouble with.
1) find the derivative of the functions:
A) f(x) = ln(e^x^2+1)
B) f(x) = x^2e^x+e^x
2) using logarithmic differentiation to find the derivative of:
f(x) = (2x^3+1) (x^2+2)^3
3)find an equation of the tangent line to the graph of:
f(x) = x^2+2 ln x at it's inflection point. (Hint: f"(x))
4) find the absolute extrema of the function:
g(x) = (2x-1)e^-x on [0,infinity]
5) find the second derivative of:
y = e^3x ln 2x
6) solve for t:
5e^-2t = 6
Answer Questioner: Ryan
Category: Calculus
Private: No
Subject: calculus
Question: I have a couple of questions regarding my calculus homework. I have already
completed most of the assignment, but these 6 problems are what I am
having trouble with.
1) find the derivative of the functions:
A) f(x) = ln(e^x^2+1)
B) f(x) = x^2e^x+e^x
2) using logarithmic differentiation to find the derivative of:
f(x) = (2x^3+1) (x^2+2)^3
3)find an equation of the tangent line to the graph of:
f(x) = x^2+2 ln x at it's inflection point. (Hint: f"(x))
4) find the absolute extrema of the function:
g(x) = (2x-1)e^-x on [0,infinity]
5) find the second derivative of:
y = e^3x ln 2x
6) solve for t:
5e^-2t = 6
..................................
Hi, Ryan,
When someone sends me a whole bunch of problems I assume he just want some hints, not complete solutions, SO:
1) find the derivative of the functions:
A) f(x) = ln(e^x^2+1)
Chain rule:
1
f' = ---------- D(e^(x^2) + 1)
e^x^2+1
and D(e^(x^2)) is another chain rule.
B) f(x) = x^2e^x+e^x
f(x) = e^x(x^2 + 1)
Now use the product rule. [Not the only way to do it, of course.]
2) using logarithmic differentiation to find the derivative of:
f(x) = (2x^3+1) (x^2+2)^3
Write y = (2x^3+1) (x^2+2)^3
Write ln y = ln[(2x^3+1) (x^2+2)^3]
Use log properties:
ln y = ln(2x^3+1) + 3 ln (x^2+2)
Now chain rules for diff:
1 dy 6x^2 6x
--- -- = ------- + ----------
y dx 2x^3 +1 x^2 + 2
Replace y by ln(...) ,multiply through, and simplify.
3)find an equation of the tangent line to the graph of:
f(x) = x^2+2 ln x at it's inflection point. (Hint: f"(x))
[No apostrophe in its, please.]
f' = 2x + 2/x
f'' = 2 - 2/x^2.
Solve and get x = 1. [Not +- 1, just 1]
y = f(1) = 1. [Remember, ln 1 = 0]
f'(1) = 2 + 2 = 4.
Put that all together.
4) find the absolute extrema of the function:
g(x) = (2x-1)e^-x on [0,infinity]
Get g'(x), using Product Rule.
Set = 0, solve.
Test solution(s) and test x = 0 for absolute max, min.
