Advertisements
Advertisements
Question
Find the derivatives of the following:
`sqrt(x^2 + y^2) = tan^-1 (y/x)`
Advertisements
Solution
`sqrt(x^2 + y^2) = tan^-1 (y/x)`
⇒ `tan sqrt(x^2+ y^2)= y/x`
Differentiating with respect to x
`sec^2 sqrt(x^2 + y^2) xx 1/2 (x^2 + y^2)^(1/2 - 1) (2x + 2y ("d"y)/("d"x)) = (x ("d"y)/(""x) - y xx 1)/x^2`
`sec^2 sqrt(x^2 + y^2) xx 1/2 (x^2 + y^2)^(- 1/2) (x + y ("d"y)/("d"x)) = (x ("d"y)/("d"x) - y)/x^2`
`(sec^2 sqrt(x^2 + y^2))/(sqrt(x^2 + y^2)) (x + y (""y)/("d"x)) = (x ("d"y)/("d"x) - y)/x^2`
`(x^2 * sec^2 sqrt(x^2 + y^2))/(sqrt(x^2 + y^2)) (x + y ("d"y)/("d"x)) = x ("d"y)/("d"x) - y`
`x^2/(sqrt(x^2 + y^2)) (1 + tan^2 sqrt(x^2 + y^2)) (x + y ("d"y)/("d"x)) = x ("d"y)/("d"x) - y`
`x^2/(sqrt(x^2 + y^2)) (1 + (y^2)/(x^2)) (x + y ("d"y)/("d"x)) = x ("d"y)/("d"x) - y`
`x^2/(sqrt(x^2 + y^2)) ((x^2 + y^2)/x^2) (x + y ("d"y)/("d"x)) = x ("d"y)/("d"x) - y`
`(x^2 + y^2)/(sqrt(x^2 + y^2)) (x + y ("d"y)/("d"x)) = x ("d"y)/("d"x) - y`
`sqrt(x^2 + y^2) (x + y ("d"y)/("d"x)) = x ("d"y)/("d"x) - y`
`x sqrt(x^2 + y^2) + y sqrt(x^2 + y^2) ("d"y)/("d"x) = x ("d"y)/("d"x) - y`
`y sqrt(x^2 + y^2) ("d"y)/("d"x) - x ("d"y)/("d"x) = - y - x sqrt(x^2 + y^2)`
`(y sqrt(x^2 + y^2) - x)("d"y)/("d"x) = -(x sqrt(x^2 + y^2) + y)`
`("d"y)/("d"x) = - ((x sqrt(x^2 + y^2) + y))/(y sqrt(x^2 + y^2) - x)`
`("d"y)/("d"x) = (x sqrt(x^2 + y^2) + y)/(x - y sqrt(x^2 + y^2))`
APPEARS IN
RELATED QUESTIONS
Find the derivatives of the following functions with respect to corresponding independent variables:
f(x) = x – 3 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 = `sinx/(1 + cosx)`
Find the derivatives of the following functions with respect to corresponding independent variables:
y = `x/(sin x + cosx)`
Find the derivatives of the following functions with respect to corresponding independent variables:
y = cosec x . cot x
Differentiate the following:
y = `"e"^sqrt(x)`
Differentiate the following:
F(x) = (x3 + 4x)7
Differentiate the following:
h(t) = `("t" - 1/"t")^(3/2)`
Differentiate the following:
y = cos (a3 + x3)
Differentiate the following:
y = (2x – 5)4 (8x2 – 5)–3
Differentiate the following:
y = tan (cos x)
Differentiate the following:
y = sin3x + cos3x
Differentiate the following:
y = `sin(tan(sqrt(sinx)))`
Find the derivatives of the following:
xy = yx
Find the derivatives of the following:
`tan^-1sqrt((1 - cos x)/(1 + cos x)`
Find the derivatives of the following:
`tan^-1 ((cos x + sin x)/(cos x - sin x))`
Find the derivatives of the following:
If y = sin–1x then find y”
Choose the correct alternative:
If y = `1/("a" - z)`, then `("d"z)/("d"y)` is
Choose the correct alternative:
The differential coefficient of `log_10 x` with respect to `log_x 10` is
Choose the correct alternative:
If f(x) = `{{:("a"x^2 - "b"",", - 1 < x < 1),(1/|x|",", "elsewhere"):}` is differentiable at x = 1, then
