Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
Given sin y = x sin(a + y) ........(1)
Differentiating with respect to x, we get
`cos ("d"y)/("d"x) = x cos("a" + y) (0 + ("d"y)/("d"x)) + sin("a" + y) * 1`
`cos y ("d"y)/("d"x) = xcos("a" + y) ("d"y)/("d"x) + sin("a" + y)`
`cos y ("d"y)/("d"x) - xcos("a" + y) ("d"y)/("d"x) = sin("a" + y)`
`("d"y)/("d"x) = (sin("a" + y))/(cosy- xcos("a" + y))` ........(2)
From equation (1) we have, x = `sin y/(sin("a" + y))`
Substituting for x in equation (2) we get
`("d"y)/("d"x) = (sin("a" + y))/(cosy - siny/(sin("a" + y)) * cos("a" + y))`
`("d"y)/("d"x) = (sin("a" + y))/((sin("a" + y) cosy - cos("a" + y) sin y)/(sin("a" + y))`
= `(sin^2("a" + y))/(sin ["a" + y - y])`
sin(A – B) = sinA cosB – cosA sin B
`("d"y)/("d"x) = (sin^2("a" + y))/sin "a"`
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 = sin x0
Differentiate the following:
y = (x2 + 4x + 6)5
Differentiate the following:
y = `(x^2 + 1) root(3)(x^2 + 2)`
Differentiate the following:
y = `x"e"^(-x^2)`
Differentiate the following:
y = sin2(cos kx)
Differentiate the following:
y = (1 + cos2)6
Find the derivatives of the following:
(cos x)log x
Find the derivatives of the following:
`x^2/"a"^2 + y^2/"b"^2` = 1
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:
`cos[2tan^-1 sqrt((1 - x)/(1 + x))]`
Find the derivatives of the following:
x = `"a" cos^3"t"` ; y = `"a" sin^3"t"`
Find the derivatives of the following:
Find the derivative with `tan^-1 ((sinx)/(1 + cos x))` with respect to `tan^-1 ((cosx)/(1 + sinx))`
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:
`"d"/("d"x) ("e"^(x + 5log x))` 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 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?
