Find the derivatives of the following functions with respect to corresponding independent variables: y = cosec x . cot x

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

y = cosec x . cot x

Sum

y = u v

⇒ y’ = uv’ + vu’

u = cosec x

⇒ u’ = – cosec x cot x

v = cot x

⇒ v’ = – cosec² x

`("d"y)/("d"x)` = (cosec x)(– cosec²x) + cot x(– coseç x cot x)

= cosec³x – cosec x cot²x

= – cosec x (cosec²x + cot²x)

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

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

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

shaalaa.com

Differentiation Rules

Is there an error in this question or solution?

Chapter 10: Differential Calculus - Differentiability and Methods of Differentiation - Exercise 10.2 [Page 160]

Chapter 10 Differential Calculus - Differentiability and Methods of Differentiation
Exercise 10.2 | Q 14 | Page 160

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

f(x) = x sin x

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

y = x sin x cos x

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

y = sin x⁰

Differentiate the following:
y = `root(3)(1 + x^3)`

Differentiate the following:

h(t) = `("t" - 1/"t")^(3/2)`

Differentiate the following:

y = `x"e"^(-x^2)`

Differentiate the following:

y = `5^((-1)/x)`

Differentiate the following:

y = `"e"^(3x)/(1 + "e"^x`

Find the derivatives of the following:
y = `x^(logx) + (logx)^x`

Find the derivatives of the following:

Find the derivative of `sin^-1 ((2x)/(1 + x^2))` with respect to `tan^-1 x`

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 y = `(cos^-1 x)^2`, prove that `(1 - x^2) ("d"^2y)/("d"x)^2 - x ("d"y)/("d"x) - 2` = 0. Hence find y₂ when x = 0