Advertisements
Advertisements
प्रश्न
Differentiate the function with respect to x.
cos x . cos 2x . cos 3x
Advertisements
उत्तर
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
संबंधित प्रश्न
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.
`(sin x)^x + sin^(-1) sqrtx`
Differentiate the function with respect to x.
`(x cos x)^x + (x sin x)^(1/x)`
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`.
if `x^m y^n = (x + y)^(m + n)`, prove that `(d^2y)/(dx^2)= 0`
Find `(dy)/(dx) , if y = sin ^(-1) [2^(x +1 )/(1+4^x)]`
Find `dy/dx` if y = xx + 5x
If y = (log x)x + xlog x, find `"dy"/"dx".`
If y = `x^(x^(x^(.^(.^.∞))`, then show that `"dy"/"dx" = y^2/(x(1 - logy).`.
If x = `(2bt)/(1 + t^2), y = a((1 - t^2)/(1 + t^2)), "show that" "dx"/"dy" = -(b^2y)/(a^2x)`.
Differentiate 3x w.r.t. logx3.
Find the second order derivatives of the following : x3.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.
Find the nth derivative of the following: log (ax + b)
If y = `25^(log_5sin_x) + 16^(log_4cos_x)` then `("d"y)/("d"x)` = ______.
If y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))]`, find `("d"y)/("d"x)`
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`
If y = `(sin x)^sin x` , then `"dy"/"dx"` = ?
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 _______.
If y = `{f(x)}^{phi(x)}`, then `dy/dx` is ______
If y = tan-1 `((1 - cos 3x)/(sin 3x))`, then `"dy"/"dx"` = ______.
`"d"/"dx" [(cos x)^(log x)]` = ______.
If y = `log ((1 - x^2)/(1 + x^2))`, then `"dy"/"dx"` is equal to ______.
`lim_("x" -> 0)(1 - "cos x")/"x"^2` is equal to ____________.
If `"y" = "e"^(1/2log (1 + "tan"^2"x")), "then" "dy"/"dx"` is equal to ____________.
If y = `x^(x^2)`, then `dy/dx` is equal to ______.
If `log_10 ((x^3 - y^3)/(x^3 + y^3))` = 2 then `dy/dx` = ______.
If `log_10 ((x^2 - y^2)/(x^2 + y^2))` = 2, then `dy/dx` is equal to ______.
If y = `9^(log_3x)`, find `dy/dx`.
Evaluate:
`int log x dx`
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 the first step in the standard procedure for \[y=[u(x)]^{v(x)}\]?
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?
