Advertisements
Advertisements
प्रश्न
Differentiate the following:
y = `sqrt(x + sqrt(x + sqrt(x)`
Advertisements
उत्तर
y = `sqrt(x + sqrt(x + sqrt(x)`
⇒ y = `[x + (x + x^(1/2))^(1/2)]^(1/2)`
y = f(g(x))
`("d"y)/("dx)` = f'(g(x)) . g'(x)
`("d"y)/("d"x) = 1/2 [x (x + x^(1/2))^(1/2)]^(1/2 - 1) xx "d"/("d"x) [x + (x + x^(1/2))^(1/2)]`
= `1/2[x + (x + x^(1/2))^(1/2)]^(- 1/2) xx [1 +1/2 (x + x^(1/2))^(1/2 - 1) xx "d"/("d"x) (x + x^(1/2))]`
= `1/2[x + (x + x^(1/2))^(1/2)]^(- 1/2) xx [1 +1/2 (x + x^(1/2))^(-1/2) xx (1 + 1/2 x^(1/2 - 1))]`
= `1/2[x + (x + x^(1/2))^(1/2)]^(- 1/2) xx [1 +1/2 (x + x^(1/2))^(-1/2) xx (1 + 1/2 x^(-1/2))]`
= `1/2[x + (x + x^(1/2))^(1/2)]^(- 1/2) [1 + 1/2 (x + x^(1/2))^(-1/2) (1 + 1/(2x^(1/2)))]`
= `1/2[x + sqrt(x + sqrt(x))]^(- 1/2) [1 + 1/(2(x + x^(1/2))^(1/2)) xx (1 + 1/(2sqrt(x)))]`
=`1/(2[x + sqrt(x + sqrt(x))]^(1/2)) xx [1 +1/(2sqrt(x sqrt(x))) xx (2sqrt(x +1))/(2sqrt(x))]`
= `1/(2sqrt(x + sqrt(x + sqrt(x)))) xx (4sqrt(x) * sqrt(x + sqrt(x)) + 2sqrt(x) + 1)/(4sqrt(x) sqrt(x + sqrt(x))`
`("d"y)/("d"x) = (4sqrt(x) * sqrt(x + sqrt(x)) + 2sqrt(x) + 1)/(8sqrt(x) * sqrt(x + sqrt(x)) * sqrt(x + sqrt(x + sqrt(x))`
APPEARS IN
संबंधित प्रश्न
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)
Find the derivatives of the following functions with respect to corresponding independent variables:
y = cosec x . cot x
Differentiate the following:
y = 4 sec 5x
Differentiate the following:
y = `5^((-1)/x)`
Differentiate the following:
y = (1 + cos2)6
Differentiate the following:
y = `"e"^(xcosx)`
Find the derivatives of the following:
y = `x^(cosx)`
Find the derivatives of the following:
y = `x^(logx) + (logx)^x`
Find the derivatives of the following:
`sqrt(x) = "e"^((x - y))`
Find the derivatives of the following:
`x^2/"a"^2 + y^2/"b"^2` = 1
Find the derivatives of the following:
tan (x + y) + tan (x – y) = x
Find the derivatives of the following:
If cos(xy) = x, show that `(-(1 + ysin(xy)))/(xsiny)`
Find the derivatives of the following:
x = `"a" cos^3"t"` ; y = `"a" sin^3"t"`
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"`
Choose the correct alternative:
x = `(1 - "t"^2)/(1 + "t"^2)`, y = `(2"t")/(1 + "t"^2)` then `("d"y)/("d"x)` is
Choose the correct alternative:
The number of points in R in which the function f(x) = |x – 1| + |x – 3| + sin x is not differentiable, is
