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Solutions to various problems related to logarithmic differentiation. It covers finding derivatives of functions g(x) and h(x) so that f(x) = g(x)h(x), simplifying natural logarithms of complex functions, and applying the chain rule to find the derivative of f(x) with respect to x. The document also includes examples with functions such as (x + 1)tan(x) and addresses false statements about differentiability.
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Read the instructions on the board.
x^2 + 1
)2 log 10 (x)
(a) Define functions g and h so that f (x) = g(x)h(x). Then find their derivatives.
(b) Simplify ln(f (x)).
(c) Differentiate ln(f (x)) with respect to x.
(d) Use the Chain Rule to show that for a differentiable function F with F (x) > 0 for all x in the domain of F , d dx
ln(F (x))
F ′(x) F (x)
(e) Use parts (c) and (d) to find f ′(x).
The work shown below is a bit inefficient. Do it more quickly.
d dx
x + 1
x + 1 (^) dxd (1) − (^1) dxd
x + 1
x + 1)^2
−^12 (x + 1)−^1 /^2 x + 1
(x + 1)^3 /^2
The following statements are false. Find a counterexample to each, and check that it is actually a counterexample.
f (x)g(x)
= f ′(x)g′(x).
