You can Download Samacheer Kalvi 12th Maths Book Solutions Guide Pdf, Tamilnadu State Board help you to revise the complete Syllabus and score more marks in your examinations.

## Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.6

Question 1.

If u(x, y) = x^{2}y + 3xy^{4}, x = e^{t} and y = sin t, find \(\frac{d u}{d t}\) and evaluate it at t = 0.

Solution:

= (2xy + 3y^{4}) (e^{t}) + (x^{2} + 12xy^{3}) (cos t)

= (2e^{t} sin t + 3 sin^{4} t) e^{t} + [e^{2t} + 12e^{t} sin^{3} t] cos t

= e^{t} [2e^{t} sin t + 3 sin^{4} t + e^{t} (cos t) + 12 sin^{3}t cos t]

at t = 0

Question 2.

Solution:

Question 3.

If w(x, y, z) = x^{2} + y^{2} + z^{2}, x = e^{t}, y = e^{t} sin t, and z = e^{t}cos t, find \(\frac{d w}{d t}\)

Solution:

w(x, y, z) = x^{2} + y^{2} + z^{2} ; x = e^{t} ; y = e^{t} sin t, z = e^{t} cos t

Question 4.

Solution:

(Replace x, y, z value)

Question 5.

If w(x, y) = 6x^{3} – 3xy + 2y^{2}, x = e^{s}, y = cos s, s ∈ R, find \(\frac{d w}{d s}\), and evaluate at s = 0.

Solution:

Question 6.

If z(x, y) = x tan^{-1} (x y), x = t^{2}, y = s e^{t}, s, t ∈ R, Find \(\frac{\partial z}{\partial \mathbf{t}}\) and \(\frac{\partial z}{\partial \mathbf{t}}\) at s = t = 1

Solution:

z (x, y) = x tan^{-1} (xy) ; x = t^{2} ; y = se^{t}

Question 7.

Let U (x, y) = e^{x} sin y, where x = st^{2}, y = s^{2}t, s, t ∈ R. Find them at s = t = 1.

Solution:

(Replace x, y values)

Question 8.

Let z(x, y) = x^{3} – 3x^{2}y^{3}, where x = se^{t}, y = se^{-t}, s, t ∈ R. Find \(\frac{\partial z}{\partial s}\) and \(\frac{\partial z}{\partial t}\)

Solution:

Question 9.

W(x,y, z) = xy + yz + zx, x = u -v, y = uv, z = u + v, u, v e R. Find \(\frac{\partial \boldsymbol{w}}{\partial \boldsymbol{u}}\), \(\frac{\partial \boldsymbol{w}}{\partial \boldsymbol{v}}\) them at \(\left(\frac{1}{2}, 1\right)\)

Solution:

### Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.6 Additional Problems

Question 1.

Suppose that z = \(y e^{x^{2}}\) where x = 2t and y = 1 – t then find \(\frac{d z}{d t}\).

Solution:

(Since x = 2t and y = 1 – t)

Question 2.

If w = x + 2y + z^{2} and x = cos t ; y = sin t ; z = t. Find \(\frac{d w}{d t}\).

Solution: