Advertisements
Advertisements
प्रश्न
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)`
Advertisements
उत्तर
u = `tan^-1 (sqrt(1 + x^2) - 1)/x`
Put x = tan θ,
u = `tan^-1 ((sqrt(1 + tan^2theta) - 1)/tan theta)`
= `tan^-1 ((sqrt(sec^2theta) - 1)/tan theta)`
= `tan^-1 ((sectheta - 1)/tantheta)`
= `tan^-1 ((1/(costheta) - 1)/((sintheta)/costheta))`
= `tan^-1 (((1 - costheta)/costheta)/((sin theta)/(costheta)))`
= `tan^-1 ((1 - cos theta)/sintheta)`
= `tan^-1 ((2 sin^2 theta/2)/(2 sin theta/2 cos theta/2))`
= `tan^-1 ((sin theta/2)/(cos theta/2))`
= `tan^-1 (tan theta/2)`
= `theta/2`
u = `1/2 tan^-1 (x)`
`("d"u)/("d"x) = 1/2 xx 1/(1 + x^2)` .......(1)
Let v = `tan^-1 (x)`
`("d"v)/("d"x) = 1/(1 + x^2)` ........(2)
From equations (1) and (2)
`(("d"u)/("d"x))/(("d"v)/("d"x)) = (1/(2(1 + x^2)))/(1/(1 + x^2))`
`("d"u)/("d"v) = 1/2`
`("d"(tan^-1 ((sqrt(1 + x^2) - 1)/x)))/("d"(tan^-1 x)) = 1/2`
APPEARS IN
संबंधित प्रश्न
Find the derivatives of the following functions with respect to corresponding independent variables:
y = ex sin x
Find the derivatives of the following functions with respect to corresponding independent variables:
y = `tan x/x`
Find the derivatives of the following functions with respect to corresponding independent variables:
y = sin x0
Find the derivatives of the following functions with respect to corresponding independent variables:
Draw the function f'(x) if f(x) = 2x2 – 5x + 3
Differentiate the following:
y = (2x – 5)4 (8x2 – 5)–3
Differentiate the following:
s(t) = `root(4)(("t"^3 + 1)/("t"^3 - 1)`
Differentiate the following:
y = tan (cos x)
Differentiate the following:
y = `sqrt(x +sqrt(x)`
Find the derivatives of the following:
`sqrt(x^2 + y^2) = tan^-1 (y/x)`
Find the derivatives of the following:
`tan^-1sqrt((1 - cos x)/(1 + cos x)`
Find the derivatives of the following:
x = `"a" cos^3"t"` ; y = `"a" sin^3"t"`
Find the derivatives of the following:
x = `(1 - "t"^2)/(1 + "t"^2)`, y = `(2"t")/(1 + "t"^2)`
Find the derivatives of the following:
Find the derivative with `tan^-1 ((sinx)/(1 + cos x))` with respect to `tan^-1 ((cosx)/(1 + sinx))`
Find the derivatives of the following:
If y = etan–1x, show that (1 + x2)y” + (2x – 1)y’ = 0
Choose the correct alternative:
If y = `1/4 u^4`, u = `2/3 x^3 + 5`, then `("d"y)/("d"x)` 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 y = `(1 - x)^2/x^2`, then `("d"y)/("d"x)` 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?
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
