Advertisements
Advertisements
Question
Differentiate the function with respect to x.
cos x . cos 2x . cos 3x
Advertisements
Solution
Let, y = cos x · cos 2x · cos 3x .....(1)
Taking logarithm of both the sides,
log y = log (cos x · cos 2x · cos 3x)
log y = log cos x + log cos 2x + log cos 3x ....[∵ log m · n = log m + log n]
Differentiating both sides with respect to x,
`1/y dy/dx = d/dx log cos x + d/dx log cos 2x + d/dx log cos 3 x`
`1/y dy/dx = 1/(cos x) d/dx cos x + 1/(cos 2x) d/dx cos 2x + 1/(cos 3x) d/dx cos 3x`
`1/y dy/dx = 1/cos x (- sin x) + 1/(cos 2x) (- sin 2x) d/dx (2x) + 1/(cos 3x) (- sin 3x) d/dx (3x)`
`1/y dy/dx = - sin/cos x - (sin 2 x)/(cos 2 x) (2) - (sin 3 x)/(cos 3 x) (3)`
`1/y dy/dx` = − tan x − 2 tan 2x − 3 tan 3x
`1/y dy/dx` = −(tan x + 2 tan 2x + 3 tan 3x)
∴ `dy/dx` = −y(tan x + 2 tan 2x + 3 tan 3x)
Putting the value of y from equation (1)
`dy/dx` = cos x · cos 2x · cos 3x (tan x + 2 tan 2x + 3 tan 3x)
APPEARS IN
RELATED QUESTIONS
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
if xx+xy+yx=ab, then find `dy/dx`.
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.
`x^(xcosx) + (x^2 + 1)/(x^2 -1)`
Find `bb(dy/dx)` for the given function:
yx = xy
If x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`.
If ey ( x +1) = 1, then show that `(d^2 y)/(dx^2) = ((dy)/(dx))^2 .`
Find `dy/dx` if y = xx + 5x
Find `(d^2y)/(dx^2)` , if y = log x
Find `"dy"/"dx"` , if `"y" = "x"^("e"^"x")`
xy = ex-y, then show that `"dy"/"dx" = ("log x")/("1 + log x")^2`
Find `"dy"/"dx"` if y = xx + 5x
If `"x"^(5/3) . "y"^(2/3) = ("x + y")^(7/3)` , the show that `"dy"/"dx" = "y"/"x"`
If `log_10((x^3 - y^3)/(x^3 + y^3))` = 2, show that `dy/dx = -(99x^2)/(101y^2)`.
If `log_5((x^4 + y^4)/(x^4 - y^4)) = 2, "show that""dy"/"dx" = (12x^3)/(13y^3)`.
If y = `x^(x^(x^(.^(.^.∞))`, then show that `"dy"/"dx" = y^2/(x(1 - logy).`.
If x = `asqrt(secθ - tanθ), y = asqrt(secθ + tanθ), "then show that" "dy"/"dx" = -y/x`.
Differentiate 3x w.r.t. logx3.
Find the second order derivatives of the following : log(logx)
If y = `25^(log_5sin_x) + 16^(log_4cos_x)` then `("d"y)/("d"x)` = ______.
lf y = `2^(x^(2^(x^(...∞))))`, then x(1 - y logx logy)`dy/dx` = ______
If y = tan-1 `((1 - cos 3x)/(sin 3x))`, then `"dy"/"dx"` = ______.
`d/dx(x^{sinx})` = ______
`"d"/"dx" [(cos x)^(log x)]` = ______.
If y = `("e"^"2x" sin x)/(x cos x), "then" "dy"/"dx" = ?`
Derivative of `log_6`x with respect 6x to is ______
`8^x/x^8`
`lim_("x" -> -2) sqrt ("x"^2 + 5 - 3)/("x" + 2)` is equal to ____________.
If `log_10 ((x^3 - y^3)/(x^3 + y^3))` = 2 then `dy/dx` = ______.
Derivative of log (sec θ + tan θ) with respect to sec θ at θ = `π/4` is ______.
Find `dy/dx`, if y = (sin x)tan x – xlog x.
Evaluate:
`int log x dx`
If \[y=x^x+x^{\frac{1}{x}}\] then \[\frac{\mathrm{d}y}{\mathrm{d}x}\] is equal to
For which type of function is logarithmic differentiation especially useful?
If \[y=[u(x)]^{v(x)}\], which expression gives \[\frac{dy}{dx}\] in logarithmic differentiation?
What is \[\frac{1}{y}\cdot\frac{dy}{dx}\] for \[y=\sqrt{\frac{(x-3)(x^2+4)}{3x^2+4x+5}}\]?
Which is the derivative of \[\sqrt{\frac{(x-3)(x^2+4)}{3x^2+4x+5}}\]?
What is \[\frac{dy}{dx}\] for \[y=x^{\sin x}\], \[x>0\]?
