Advertisements
Advertisements
Question
Find the derivatives of the following:
`tan^-1 = ((6x)/(1 - 9x^2))`
Advertisements
Solution
Let y = `tan^-1 ((6x)/(1 - 9x^2))`
y = `tan^-1 ((2 xx 3x)/(1 - (3x)^2))` ........(1)
Put 3x = tan θ
(1) ⇒ y = `tan^-1 ((2 xx tan theta)/(1 - tan^2 theta))`
y = `tan^-1 (tan 2theta)`
y = 2θ
`("d"y)/("d"theta) = 2 xx 1` = 2 ........(2)
3x = tan θ
`3 ("d"x)/("d"theta) = sec^2theta`
`3 ("d"x)/("d"theta) = 1 + tan^2 theta`
`("d"x)/("d"theta) = 1/3 (1 + (3x)^2)`
`("d"x)/("d"theta) = 1/3 (1 + 9x^2)` .........(3)
Using equation (2) and (3) we have
`(("d"y)/("d"theta))/(("d"x)/("d"theta)) = 2/(1/3 (1 + 9x^2))`
`("d"y)/("d"x) = 6/(1 + 9x^2)`
APPEARS IN
RELATED QUESTIONS
Find the derivatives of the following functions with respect to corresponding independent variables:
y = sin x + cos x
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 = `tan x/x`
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 = e-x . log x
Differentiate the following:
y = `sqrt(x +sqrt(x)`
Differentiate the following:
y = `"e"^(xcosx)`
Find the derivatives of the following:
xy = yx
Find the derivatives of the following:
tan (x + y) + tan (x – y) = x
Find the derivatives of the following:
`tan^-1sqrt((1 - cos x)/(1 + cos x)`
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 of sin x2 with respect to x2
Find the derivatives of the following:
Find the derivative of `sin^-1 ((2x)/(1 + x^2))` with respect to `tan^-1 x`
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:
If the derivative of (ax – 5)e3x at x = 0 is – 13, then the value of a 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) = `{{:(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?
Choose the correct alternative:
If f(x) = `{{:("a"x^2 - "b"",", - 1 < x < 1),(1/|x|",", "elsewhere"):}` is differentiable at x = 1, then
