First, it's recommended to obtain a formula for the
n
th derivative of
cos
x
.
To do this, usually it is needed to continually differentiate until you notice a pattern.
So we will begin by taking the first derivative:
d
y
d
x
=
−
sin
x
Next, the second derivative:
d
2
y
d
x
2
=
−
cos
x
And the third derivative:
d
3
y
d
x
3
=
sin
x
The fourth:
d
4
y
d
x
4
=
cos
x
There - we've arrived back at
cos
x
. Since this was our original function, differentiating again will just give us the first derivative, and so on. So, we can deduce that the
n
th derivative is periodic.
Now the problem is putting this pattern into a formula. At first it might look like there's no mathematically explainable pattern - we have a negative, then a negative, then a positive, then a positive, meanwhile flipping from sine to cosine - but when you graph these successive functions, it's easy to see that each graph is the previous derivative, but shifted to the left by
π
2
.
What do I mean? Well,
−
sin
x
is the same thing as
cos
(
x
+
π
2
)
. And
−
cos
x
is the same thing as
cos
(
x
+
π
)
.
So there's our formula:
f
n
(
x
)
=
cos
(
x
+
π
n
2
)
Now, if we substitute
n
=
50
, we obtain:
f
50
(
x
)
=
cos
(
x
+
25
π
)
Since cosine itself is periodic, we can divide the
Answers & Comments
Answer:
First, it's recommended to obtain a formula for the
n
th derivative of
cos
x
.
To do this, usually it is needed to continually differentiate until you notice a pattern.
So we will begin by taking the first derivative:
d
y
d
x
=
−
sin
x
Next, the second derivative:
d
2
y
d
x
2
=
−
cos
x
And the third derivative:
d
3
y
d
x
3
=
sin
x
The fourth:
d
4
y
d
x
4
=
cos
x
There - we've arrived back at
cos
x
. Since this was our original function, differentiating again will just give us the first derivative, and so on. So, we can deduce that the
n
th derivative is periodic.
Now the problem is putting this pattern into a formula. At first it might look like there's no mathematically explainable pattern - we have a negative, then a negative, then a positive, then a positive, meanwhile flipping from sine to cosine - but when you graph these successive functions, it's easy to see that each graph is the previous derivative, but shifted to the left by
π
2
.
What do I mean? Well,
−
sin
x
is the same thing as
cos
(
x
+
π
2
)
. And
−
cos
x
is the same thing as
cos
(
x
+
π
)
.
So there's our formula:
f
n
(
x
)
=
cos
(
x
+
π
n
2
)
Now, if we substitute
n
=
50
, we obtain:
f
50
(
x
)
=
cos
(
x
+
25
π
)
Since cosine itself is periodic, we can divide the
25
by
2
and leave the remainder next to the
π
:
f
50
(
x
)
=
cos
(
x
+
π
)
Which is the same thing as:
f
50
(
x
)
=
−
cos
x