Question

find the derivative of X ^ X + cos x to the power of x​

Answer 1

Answer:
d
y
d
x
=
x
cos
x
(
−
sin
x
ln
x
+
cos
x
x
)
Explanation:
y
=
x
cos
x
Take the natural logarithm of both sides.
ln
y
=
ln
(
x
cos
x
)
Use the logarithm law for powers, which states that
log
a
n
=
n
log
a
ln
y
=
cos
x
ln
x
Use the product rule to differentiate the right hand side.
d
d
x
(
cos
x
)
=
−
sin
x
and
d
d
x
(
ln
x
)
.
1
y
(
d
y
d
x
)
=
−
sin
x
(
ln
x
)
+
cos
x
(
1
x
)
1
y
(
d
y
d
x
)
=
−
sin
x
ln
x
+
cos
x
x
d
y
d
x
=
−
sin
x
ln
x
+
cos
x
x
1
y
d
y
d
x
=
x
cos
x
(
−
sin
x
ln
x
+
cos
x
x
)
Hopefully
Answer:
d
d
x
x
cos
x
=
x
cos
x
(
cos
x
x
−
sin
x
ln
x
)
Explanation:
You can write:
x
cos
x
=
(
e
ln
x
)
cos
x
=
e
ln
x
cos
x
so:
d
d
x
x
cos
x
=
d
d
x
(
e
ln
x
cos
x
)
using the chain rule:
d
d
x
x
cos
x
=
e
ln
x
cos
x
d
d
x
(
ln
x
cos
x
)
then the product rule:
d
d
x
x
cos
x
=
x
cos
x
(
cos
x
x
−
sin
x
ln
x
)