differentiate y=sinx/x²
Answers
Answer:
We have
y
=
sin
x
x
2
We can differentiate it by two methods:
Method 1:
We will use the quotient rule, there is a way I like to remember it:
Denominator same, differentiation of numerator. Minus numerator same, differentiation of denominator whole divided by denominator squared
Differentiating with respect to x:
d
y
d
x
=
x
2
⋅
cos
x
−
sin
x
⋅
2
x
(
x
2
)
2
d
y
d
x
=
x
2
⋅
cos
x
−
sin
x
⋅
2
x
x
4
d
y
d
x
=
x
2
⋅
cos
x
x
4
−
sin
x
⋅
2
x
x
4
d
y
d
x
=
cos
x
x
2
−
2
sin
x
x
3
Method 2:
Simplifying:
y
=
sin
x
⋅
x
−
2
We can now apply the product rule, there is a way I like to remember it:
First function same, differentiation of second function. Plus second function same, differentiation of first function
Differentiating with respect to x:
d
y
d
x
=
sin
x
⋅
(
−
2
)
x
−
3
+
x
−
2
⋅
cos
x
d
y
d
x
=
−
2
sin
x
⋅
x
−
3
+
x
−
2
⋅
cos
x
Rearranging the terms, we get:
d
y
d
x
=
x
−
2
⋅
cos
x
−
2
sin
x
⋅
x
−
3
d
y
d
x
=
cos
x
x
2
−
2
sin
x
x
3
I hope it helps!