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The Derivative of a Composite Function Involving y log a x
If y log , a 1 , then dy
dx f ¿1x2
a f 1x2, a 7 0
f 1x2 ln a .
Exercise
PART A
dy
1. Determine for each function.
dx
a. y log 5 x d. y 3 log 7 x
b. y log 3 x e. y 1log x2
c. y 2 log 4 x f. y 3 log 6 x
2. Determine the derivative of each function.
a.
b.
c.
d.
e.
f.
y log 3 1x 22
y log 8 12x2
y 3 log 3 12x 32
y log 10 15 2x2
y log 8 12x 62
y log 7 1x 2 x 12
PART B
3. a. If f 1t2 log 2 Q t 1 evaluate f ¿132.
2t 7 R,
b. If h1t2 log 3 1log 2 1t22, determine h¿182.
4. Differentiate.
a. y log 10 a 1 x
1 x b
b. y log 2 x 2 3x
c. y 2 log 3 15 x 2 log 3 14 x 2
d. y 3 x log 3 x
e. y 2x log 4 x
f. y log 513x 2 2
x 1
5. Determine the equation of the tangent
to the curve y x log x at x 10. Graph
the function and the tangent.
6. Explain why the derivative of y log a kx,
dy
k 7 0, is for any constant k.
dx 1
x ln a
7. Determine the equation of the tangent to the
curve y 10 2x9 log 10 1x 2 3x2 at x 5.
8. A particle’s distance, in metres, from a
fixed point at time, t, in seconds is given by
s1t2 t log 6 1t 12, t 0. Is the distance
increasing or decreasing at t 15? How do
you know?
PART C
9. a. Determine the equation of the tangent
to y log 3 x at the point (9, 2).
b. Graph the function and include any
asymptotes.
c. Will this tangent line intersect any
asymptotes? Explain.
10. Determine the domain, critical numbers, and
intervals of increase and decrease of
f 1x2 ln1x 2 42.
11. Do the graphs of either of these functions
have points of inflection? Justify your
answers with supporting calculations.
a. y x ln x
b. y 3 2 log x
12. Determine whether the slope of the graph of
y 3 x at the point 10, 12 is greater than the
slope of the graph of y log 3 x at the point
11, 02. Include graphs with your solution.
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CALCULUS APPENDIX—THE DERIVATIVES OF GENERAL LOGARITHMIC FUNCTIONS
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