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)`
Differentiate the function with respect to x.
`sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5)))`
Differentiate the function with respect to x.
(x + 3)2 . (x + 4)3 . (x + 5)4
Differentiate the function with respect to x.
(log x)x + xlog x
Differentiate the function with respect to x.
xsin x + (sin x)cos x
Differentiate the function with respect to x.
`x^(xcosx) + (x^2 + 1)/(x^2 -1)`
Find `bb(dy/dx)` for the given function:
xy + yx = 1
Find the derivative of the function given by f(x) = (1 + x) (1 + x2) (1 + x4) (1 + x8) and hence find f′(1).
Differentiate (x2 – 5x + 8) (x3 + 7x + 9) in three ways mentioned below:
- By using the product rule.
- By expanding the product to obtain a single polynomial.
- By logarithmic differentiation.
Do they all give the same answer?
If x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`.
Find `(dy)/(dx) , if y = sin ^(-1) [2^(x +1 )/(1+4^x)]`
Evaluate
`int 1/(16 - 9x^2) dx`
If `"x"^(5/3) . "y"^(2/3) = ("x + y")^(7/3)` , the show that `"dy"/"dx" = "y"/"x"`
Solve the following differential equation: (3xy + y2) dx + (x2 + xy) dy = 0
If `(sin "x")^"y" = "x" + "y", "find" (d"y")/(d"x")`
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 = 2cos4(t + 3), y = 3sin4(t + 3), show that `"dy"/"dx" = -sqrt((3y)/(2x)`.
If x = log(1 + t2), y = t – tan–1t,show that `"dy"/"dx" = sqrt(e^x - 1)/(2)`.
Find the second order derivatives of the following : log(logx)
If y = `log(x + sqrt(x^2 + a^2))^m`, show that `(x^2 + a^2)(d^2y)/(dx^2) + x "d"/"dx"` = 0.
If y = log (log 2x), show that xy2 + y1 (1 + xy1) = 0.
Find the nth derivative of the following : log (2x + 3)
If f(x) = logx (log x) then f'(e) is ______
If y = `log[4^(2x)((x^2 + 5)/sqrt(2x^3 - 4))^(3/2)]`, find `("d"y)/("d"x)`
If x7 . y5 = (x + y)12, show that `("d"y)/("d"x) = y/x`
The rate at which the metal cools in moving air is proportional to the difference of temperatures between the metal and air. If the air temperature is 290 K and the metal temperature drops from 370 K to 330 K in 1 O min, then the time required to drop the temperature upto 295 K.
Derivative of loge2 (logx) with respect to x is _______.
`"d"/"dx" [(cos x)^(log x)]` = ______.
If y = `("e"^"2x" sin x)/(x cos x), "then" "dy"/"dx" = ?`
If `"f" ("x") = sqrt (1 + "cos"^2 ("x"^2)), "then the value of f'" (sqrtpi/2)` is ____________.
The derivative of x2x w.r.t. x is ______.
Find `dy/dx`, if y = (sin x)tan x – xlog x.
If y = `log(x + sqrt(x^2 + 4))`, show that `dy/dx = 1/sqrt(x^2 + 4)`
Find the derivative of `y = log x + 1/x` with respect to x.
If xy = yx, then find `dy/dx`
