Question 1. Differentiate x3 sin x with respect to x.
Solution:
We have,
=> y = x3 sin x
On differentiating both sides with respect to x, we get,
On using product rule, we get,
=
= sinx (3x2) + x3 (cosx)
= 3x2 sinx + x3 cosx
= x2 (3 sinx + x cos x)
Question 2. Differentiate x3 ex with respect to x.
Solution:
We have,
=> y = x3 ex
On differentiating both sides with respect to x, we get,
On using product rule, we get,
=
= ex (3x2) + x3 (ex)
= 3x2 ex + x3 ex
= x2 ex (3 + x)
Question 3. Differentiate x2 ex log x with respect to x.
Solution:
We have,
=> y = x2 ex log x
On differentiating both sides with respect to x, we get,
On using product rule, we get,
=
On using product rule again in the second part of the expression, we get,
=
=
=
=
Question 4. Differentiate xn tan x with respect to x.
Solution:
We have,
=> y = xn tan x
On differentiating both sides with respect to x, we get,
On using product rule, we get,
=
=
=
=
Question 5. Differentiate xn loga x with respect to x.
Solution:
We have,
=> y = xn loga x
On differentiating both sides with respect to x, we get,
On using product rule, we get,
=
=
=
=
=
Question 6. Differentiate (x3+x2+1)sinx with respect to x.
Solution:
We have,
=> y =
On differentiating both sides with respect to x, we get,
On using product rule, we get,
=
=
=
Question 7. Differentiate sin x cos x with respect to x.
Solution:
We have,
=> y = sin x cos x
On differentiating both sides with respect to x, we get,
On using product rule, we get,
=
= cos x (cos x) − sin x (−sin x)
= cos2 x − sin2 x
= cos2 x − (1 − cos2 x)
= cos2 x − 1 + cos2 x
= 2 cos2 x − 1
= cos 2x
Question 8. Differentiate
Solution:
We have,
=> y =
=> y =
On differentiating both sides with respect to x, we get,
On using product rule, we get,
=
On using product rule again in the second part of the expression, we get,
=
=
=
=
Question 9. Differentiate x2 sin x log x with respect to x.
Solution:
We have,
=> y = x2 sin x log x
On differentiating both sides with respect to x, we get,
On using product rule, we get,
=
On using product rule again in the second part of the expression, we get,
=
=
=
Question 10. Differentiate x5 ex + x6 log x with respect to x.
Solution:
We have,
=> y = x5 ex + x6 log x
On differentiating both sides with respect to x, we get,
On using chain rule, we get,
=
On using product rule, we get,
=
=
=
=