If y = {log (x + √x² + 1)², show that (1 + x²) .

If y = (tan ^{– 1} x)², then prove that (1 – x2)² y₂ + 2x (1 + x²) y₁ = 2

If y = cot x show that .

Find , where y = log .

If y = e^x(sin x + cos x) prove that .

If y = e^x (sin x + cos x) Prove that

If y = cos ^{– 1} x, find in terms of y alone.

If , prove that

If , show that .

If x = 2 cos t – cos 2t, y = 2 sin t – sin 2t, find .

If x = 4z² + 5, y = 6z² + 7z + 3, find .

If y = log (1 + cos x), prove that

If y = sin (log x), prove that

If , prove that .

If, prove that .

If , then show that

If , then find the value of .

If x = a sin t and , find .

If x = a (cos t + t sin t) and y = a (sin t – t cos t), then find the value of at

If , y = a sin t, evaluate at

If x = a (cos 2t + 2t sin 2t) and y = a (sin 2t – 2t cos 2t), then find .

If x = 3 cot t – 2 cos³ t, y = 3 sin t – 2 sin³ t, find .

If x = a sin t – b cos t, y = a cos t + b sin t, prove that .

Find A and B so that y = A sin 3x + Bcos 3x satisfies the equation

If y = A e ^{– kt} cos (pt + c), prove that , where n² = p² + k².

If , prove that

If y = a {x + √x² + 1}ⁿ + b{x – √x² + 1} ^{– n}, prove that (x² – 1) .

Find the second order derivatives of each of the following functions:

x³ + tan x

Find the second order derivatives of each of the following functions:

sin (log x)

Find the second order derivatives of each of the following functions:

log (sin x)

Find the second order derivatives of each of the following functions:

e^x sin 5x

Find the second order derivatives of each of the following functions:

e^6x cos 3x

Find the second order derivatives of each of the following functions:

x³ log x

Find the second order derivatives of each of the following functions:

tan^-1 x

Find the second order derivatives of each of the following functions:

x cos x

Find the second order derivatives of each of the following functions:

log (log x)

If y = x + tan x, show that: cos²

If y = x³ log x, prove that .

If y = log (sin x), prove that: cos x cose³ x.

If y = 2 sin x + 3 cos x, show that:

If y = , show that .

If x = a sec θ, y = b tan θ, prove that .

If x =a (cos θ + θ sin θ), y=a (sin θ – θ cos θ) prove that

(sin θ + θ cosθ) and .

If y = e^x cosx, prove that

If x = a cos θ , y = b sin θ, show that .

If x = a (1 – cos ³θ), y = a sin ³ θ, Prove that .

If x = a (θ + sin θ), y = a (1+ cos θ), prove that .

If x = a (θ – sin θ), y = a (1 + cos θ) find .

If x = a (1 – cos θ), y =a (θ + sin θ), prove that

If x = a (1 + cos θ), y = a (θ+ sinθ) Prove that .

If x = cos θ, y = sin³θ. Prove that

If y = sin (sin x), prove that :

If y = (sin^–1 x)², prove that: (1–x²) y₂ – xy₁– 2=0

If y = (sin^–1 x)², prove that: (1–x²) y₂–xy₁–2=0

If y =e^tan–1x, Prove that: (1+x²)y₂+(2x–1)y₁=0

If y = 3 cos (log x) + 4 sin (log x), prove that: x²y₂+xy₁+ y =0.

If y=e^2x(ax + b), show that y₂–4y₁+4y = 0.

If x = sin, show that (1–x²)y₂–xy₁–a² y = 0

If log y = tan^–1 X, show that : (1+x²)y₂+(2x–1) y₁=0.

Write the correct alternative in the following:

If x = a cos nt – b sin nt, then is

Write the correct alternative in the following:

If x = at², y = 2at, then

Write the correct alternative in the following:

If y = axⁿ⁺¹ + b x^–n, then

Write the correct alternative in the following:

Write the correct alternative in the following:

If x = t², y = t³, then

Write the correct alternative in the following:

If y = a + bx², a, b arbitrary constants, then

Write the correct alternative in the following:

If f(x) = (cos x + i sinx) (cos 2x + i sin 2x) (cos 3x + i sin 3x) …. (cos nx + i sin nx) and f(1) = 1, then f’’ (1) is equal to

Write the correct alternative in the following:

If y = a sin mx + b cos mx, then is equal to

Write the correct alternative in the following:

If then (1 – x²) f’ (x)– xf(x) =

Write the correct alternative in the following:

If then

Write the correct alternative in the following:

Let f(x) be a polynomial. Then, the second order derivative of f(e^x) is

Write the correct alternative in the following:

If y = a cos (log_e x) + b sin (log_e x), then x² y₂ + xy₁ =

Write the correct alternative in the following:

If x = 2at, y = at², where a is a constant, then

Write the correct alternative in the following:

If x = f(t) and y = g(t), then is equal to

Write the correct alternative in the following:

If y = sin (m sin^–1 x), then (1 – x²) y₂ – xy₁ is equal to

Write the correct alternative in the following:

If y = (sin^–1 x)², then (1 – x²) y₂ is equal to

Write the correct alternative in the following:

If y = e^{tan x}, then (cos² x)y₂ =

Write the correct alternative in the following:

If a > b > 0, then

Write the correct alternative in the following:

If then (2xy₁ + y)y₃ =

Write the correct alternative in the following:

If then x³ y₂ =

Write the correct alternative in the following:

If x = f(t) cos t – f’(t) sin t and y = f(t) sin t + f’(t) cos t, then

Write the correct alternative in the following:

If y^1/n + y^–1/n = 2x, then (x² – 1)y₂ + xy₁ =

Write the correct alternative in the following:

If

Then the value of a_r, 0 < r ≤ n, is equal to

Write the correct alternative in the following:

If y = x^n–1 log x, then x² y₂ + (3 – 2n) xy₁ is equal to

Write the correct alternative in the following:

If xy – log_e y = 1 satisfies the equation x(yy₂ + y₁²) – y₂ + λyy₁ = 0, then λ =

Write the correct alternative in the following:

If y² = ax² + bx + c, then y³is

If x = a cos nt – b sin nt and then find the value of λ.

If x = t² and y = t³, where a is a constant, then find

If x = 2at, y = at², where a is a constant, then find

If x = f(t) and y = g(t), then write the value of

If y = 1 – x + then write in terms of y.

If y = x + e^x, find

If y = |x – x²|, then find

If y = |log_e x|, find