Differentiate the following: y = sin-1(1-x21+x2)

Differentiate the following:

y = `sin^-1 ((1 - x^2)/(1 + x^2))`

Sum

y = `sin^-1 ((1 - x^2)/(1 + x^2))`

y = f(g(x)

`("d"y)/("d"x)` = f'(g(x)) . g'(x)

`("d"y)/("d"x) = 1/sqrt(1 - ((1 - x^2)/(1 + x^2))^2) xx "d"/("d"x) ((1 - x^2)/(1 + x^2))`

= `1/sqrt(((1 + x^2)^2 - (1 - x^2)^2)/(1 + x^2)^2) xx ((1 + x^2)(- 2x) - (1 - x^2)(2x))/(1 + x^2)^2`

= `1/sqrt((1 + 2x^2 + x^4 - (1 - 2x^2 + x^4))/((1 + x^2))) xx (-2x - 2x^3 - 2x + 2x^3)/(1 + x^2)^2`

= `(1 + x^2)/sqrt(1 + 2x^2 + x^4 - 1 + 2x^2 - x^4) xx (-4x)/(1 + x^2)^2`

= `(-4x)/(sqrt(4x^2) (1 + x^2))`

`("d"y)/("d"x) = (-4x)/(2x(1 + x^2))`

= `- 2/(1 + x^2)`

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Chapter 10: Differential Calculus - Differentiability and Methods of Differentiation - Exercise 10.3 [Page 164]

Chapter 10 Differential Calculus - Differentiability and Methods of Differentiation
Exercise 10.3 | Q 30 | Page 164

Find the derivatives of the following functions with respect to corresponding independent variables:

g(t) = t³ cos t

Find the derivatives of the following functions with respect to corresponding independent variables:

y = `sinx/x^2`

Find the derivatives of the following functions with respect to corresponding independent variables:

y = tan θ (sin θ + cos θ)

Differentiate the following:
y = tan 3x

Differentiate the following:
y = cos (tan x)

Differentiate the following:
y = `"e"^sqrt(x)`

Differentiate the following:
y = sin²(cos kx)

Find the derivatives of the following:

x = a (cos t + t sin t); y = a (sin t – t cos t)

Find the derivatives of the following:

If u = `tan^-1 (sqrt(1 + x^2) - 1)/x` and v = `tan^-1 x`, find `("d"u)/("d"v)`

Find the derivatives of the following:

If y = `(sin^-1 x)/sqrt(1 - x^2)`, show that (1 – x²)y₂ – 3xy₁ – y = 0

Find the derivatives of the following:

If x = a(θ + sin θ), y = a(1 – cos θ) then prove that at θ = `pi/2`, yⁿ = `1/"a"`

Find the derivatives of the following:

If sin y = x sin(a + y), the prove that `("d"y)/("d"x) = (sin^2("a" + y))/sin"a"`, a ≠ nπ