## Rajasthan Board RBSE Class 12 Maths Chapter 7 Differentiation Ex 7.5

Question 1.

(i) y = x^{3} + tan x

(ii) y = x^{2} + 3x + 2

(iii) y = x cos x

(iv) y = 2 sin x + 3 cos x

(v) y = e^{-x} cos x

(vi) y = a sin x – b cos x

Solution:

(i) y = x^{3} + tan x

Diff. w.r.t. x on both

Again diff. w.r.t. x on both sides

(ii) y = x^{2} + 3x + 2

Diff. w.r.t. x on both sides,

(iii) y = x cos x

Diff. w.r.t. x on both sides,

(iv) y = 2 sin x + 3 cos x

Diff. w.r.t. x on both sides,

(v) y = e^{-ex} cos x

Diff. w.r.t. x on both sides,

(vi) y = a sin x – b cos x

Diff. w.r.t. x on both sides,

Question 2.

If y = a sin x + b cos x, then prove that:

Solution:

Given

y = a sin x + b cos x …..(i)

Diff. w.r.t. x on both sides,

Question 3.

If y = sec x + tan x, then prove that:

Solution:

Given y = sec x + tan x

Diff. w.r.t. x on both sides,

Question 4.

If.y = a cos nx + b sin nx, then prove that:

Solution:

Given

y = A cos nx + b sin nx

Diff. w.r.t. x on both sides,

Question 5.

If x = a cos^{3} θ, y = a sin^{3} θ, then find \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \) at θ = \(\frac { \pi }{ 4 } \)

Solution:

Given y = a sin^{3} θ

and x = a cos^{3} θ

Diff. w.r.t. θ on both sides,

Question 6.

If x^{3} + y^{3} – 3axy = 0, then prove that

Solution:

Given,

x^{3} + y^{3} – 3axy = 0

Diff. w.r.t x on both sides,

Question 7.

If y = sin^{-1} x, then prove that:

Solution:

Given

y = sin^{-1 }x

Diff. w.r.t. x on both sides

Question 8.

If y = (sin^{-1} x)^{2}, then prove that:

Solution:

Given y = (sin^{-1} x)^{2}

Diff. w.r.t. x on both sides,

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