Advertisements
Advertisements
Question
Differentiate the following:
y = `sqrt(x + sqrt(x + sqrt(x)`
Advertisements
Solution
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
RELATED QUESTIONS
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 = (x2 + 5) log(1 + x) e–3x
Differentiate the following:
y = `root(3)(1 + x^3)`
Differentiate the following:
y = 4 sec 5x
Differentiate the following:
y = `(x^2 + 1) root(3)(x^2 + 2)`
Differentiate the following:
y = `5^((-1)/x)`
Differentiate the following:
y = sin3x + cos3x
Differentiate the following:
y = (1 + cos2)6
Find the derivatives of the following:
`sqrt(x^2 + y^2) = tan^-1 (y/x)`
Find the derivatives of the following:
x = `"a" cos^3"t"` ; y = `"a" sin^3"t"`
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
Choose the correct alternative:
`"d"/("d"x) (2/pi sin x^circ)` is
Choose the correct alternative:
If y = cos (sin x2), then `("d"y)/("d"x)` at x = `sqrt(pi/2)` is
Choose the correct alternative:
If f(x) = x tan-1x then f'(1) is
Choose the correct alternative:
If the derivative of (ax – 5)e3x at x = 0 is – 13, then the value of a is
Choose the correct alternative:
If x = a sin θ and y = b cos θ, then `("d"^2y)/("d"x^2)` is
Choose the correct alternative:
If f(x) = `{{:(x - 5, "if" x ≤ 1),(4x^2 - 9, "if" 1 < x < 2),(3x + 4, "if" x ≥ 2):}` , then the right hand derivative of f(x) at x = 2 is
Choose the correct alternative:
If f(x) = `{{:(2"a" - x, "for" - "a" < x < "a"),(3x - 2"a", "for" x ≥ "a"):}` , then which one of the following is true?
