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Mathematics
Mathematics
Problem
Consider the function
g
(
x
)
=
4
x
3
−
6
x
+
5
sin
x
−
2
cos
x
g(x)=4x^{3}-\dfrac{6}{x}+5\sin x-2\cos x
g
(
x
)
=
4
x
3
−
x
6
+
5
sin
x
−
2
cos
x
. a. (i) Find
g
′
(
x
)
g'(x)
g
′
(
x
)
and simplify each term. b. (ii) Show that
g
′
(
x
)
=
12
x
2
+
6
x
2
+
5
cos
x
+
2
sin
x
g'(x)=12x^{2}+\dfrac{6}{x^{2}}+5\cos x+2\sin x
g
′
(
x
)
=
12
x
2
+
x
2
6
+
5
cos
x
+
2
sin
x
. c. (iii) Using your result from part (ii), calculate
g
′
(
π
2
)
g'(\dfrac{\pi}{2})
g
′
(
2
π
)
exactly.
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