Advertisements
Advertisements
प्रश्न
Find `bb(dy/dx)` for the given function:
(cos x)y = (cos y)x
Advertisements
उत्तर
Given, (cos x)y = (cos y)x
Taking logarithm of both sides,
log (cos x)y = log (cos y)x
y log cos x = x log cos y
Differentiating both sides with respect to x,
`y d/dx log cos x + log cos x d/dx (y)= x d/dx log cos y + log cos y d/dx (x)`
⇒ `y * 1/(cos x) d/dx cos x + log cos x * dy/dx= x * 1/(cos y) d/dx cos y + log cos y xx 1`
⇒ `y * 1/(cos x) (- sin x) + log cos x. dy/dx = x 1/(cos y) (-sin y) dy/dx + log cos y`
⇒ `-y tan x + log cos x dy/dx = - x tan y dy/dx + log cos y`
⇒ `log cos x dy/dx + x tan y dy/dx = log cos y + y tan x`
⇒ `dy/dx (log cos x + x tan y) = log cos y + y tan x`
`therefore dy/dx = (log cos y + y tan x)/ (log cos x + x tan y)`
APPEARS IN
संबंधित प्रश्न
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
Differentiate the function with respect to x.
cos x . cos 2x . cos 3x
Differentiate the function with respect to x.
xx − 2sin x
Differentiate the function with respect to x.
(log x)x + xlog x
Differentiate the function with respect to x.
`(sin x)^x + sin^(-1) sqrtx`
Differentiate the function with respect to x.
xsin x + (sin x)cos x
If u, v and w are functions of x, then show that `d/dx(u.v.w) = (du)/dx v.w + u. (dv)/dx.w + u.v. (dw)/dx` in two ways-first by repeated application of product rule, second by logarithmic differentiation.
if `x^m y^n = (x + y)^(m + n)`, prove that `(d^2y)/(dx^2)= 0`
Solve the following differential equation: (3xy + y2) dx + (x2 + xy) dy = 0
If xy = ex–y, then show that `"dy"/"dx" = logx/(1 + logx)^2`.
`"If" y = sqrt(logx + sqrt(log x + sqrt(log x + ... ∞))), "then show that" dy/dx = (1)/(x(2y - 1).`
If x = esin3t, y = ecos3t, then show that `dy/dx = -(ylogx)/(xlogy)`.
If x = a cos3t, y = a sin3t, show that `"dy"/"dx" = -(y/x)^(1/3)`.
If x = sin–1(et), y = `sqrt(1 - e^(2t)), "show that" sin x + dy/dx` = 0
Differentiate 3x w.r.t. logx3.
If y = log (log 2x), show that xy2 + y1 (1 + xy1) = 0.
Find the nth derivative of the following: log (ax + b)
Find the nth derivative of the following : log (2x + 3)
If y = `25^(log_5sin_x) + 16^(log_4cos_x)` then `("d"y)/("d"x)` = ______.
If y = log [cos(x5)] then find `("d"y)/("d"x)`
If y = `{f(x)}^{phi(x)}`, then `dy/dx` is ______
`"d"/"dx" [(cos x)^(log x)]` = ______.
`log (x + sqrt(x^2 + "a"))`
If `"y" = "e"^(1/2log (1 + "tan"^2"x")), "then" "dy"/"dx"` is equal to ____________.
If y `= "e"^(3"x" + 7), "then the value" |("dy")/("dx")|_("x" = 0)` is ____________.
Given f(x) = `log((1 + x)/(1 - x))` and g(x) = `(3x + x^3)/(1 + 3x^2)`, then fog(x) equals
If y = `9^(log_3x)`, find `dy/dx`.
Find the derivative of `y = log x + 1/x` with respect to x.
If \[y=x^x+x^{\frac{1}{x}}\] then \[\frac{\mathrm{d}y}{\mathrm{d}x}\] is equal to
For \[y=x^{\sin x}\], \[x>0\], what equation results after taking logarithm on both sides?
After taking logarithms in logarithmic differentiation, which rules are used to simplify products, quotients and powers before differentiation?
Which derivative correctly represents differentiating \[\ln y\] carefully?
