Advertisements
Advertisements
प्रश्न
Differentiate the function with respect to x.
(x + 3)2 . (x + 4)3 . (x + 5)4
Advertisements
उत्तर
Let, y = (x + 3)2 · (x + 4)3 · (x + 5)4
Taking logarithm of both sides,
log y = log [(x + 3)2 · (x + 4)3 · (x + 5)4]
= log (x + 3)2 + log (x + 4)3 + log (x + 5)4 ...[∵ log mn = log m + log n]
= 2 log (x + 3) + 3 log (x + 4) + 4 log (x + 5) ...[∵ log mn = n log m]
Differentiating both sides with respect to x,
`1/y dy/dx = 2 d/dx log (x + 3) + 3 d/dx log (x + 4) + 4 d/dx log (x + 5)`
`1/y dy/dx = 2 * 1/(x + 3) d/dx (x + 3) + 3 xx 1/(x+ 4) d/dx (x + 4) + 4 xx 1/(x + 5) d/dx (x + 5)`
`1/y dy/dx = (2(1 + 0))/(x + 3) + (3(1 + 0))/("x" + 4) + (4(1 + 0))/(x + 5)`
`dy/dx = y [2/(x + 3) + 3/(x + 4) + 4/(x + 5)]`
`= y [(2 (x + 4) (x + 5) + 3 (x + 5) + 4 (x + 3) (x + 4))/((x + 3) (x + 4) (x + 5))]`
`= (x + 3)^2 (x + 4)^3 (x + 5)^4 xx [(2 (x^2 + 9x + 20) + 3(x^2 + 8x + 15) + 4 (x^2 + 7x + 12))/((x + 3) (x + 4) (x + 5))]`
`⇒ dy/dx= (x + 3) (x + 4)^2 (x + 5)^3 [9x^2 + 70x + 133]`
APPEARS IN
संबंधित प्रश्न
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^(xcosx) + (x^2 + 1)/(x^2 -1)`
if `x^m y^n = (x + y)^(m + n)`, prove that `(d^2y)/(dx^2)= 0`
If `y = sin^-1 x + cos^-1 x , "find" dy/dx`
Find `(dy)/(dx) , if y = sin ^(-1) [2^(x +1 )/(1+4^x)]`
xy = ex-y, then show that `"dy"/"dx" = ("log x")/("1 + log x")^2`
Differentiate : log (1 + x2) w.r.t. cot-1 x.
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 `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 = 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
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
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.
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 x7 . y5 = (x + y)12, show that `("d"y)/("d"x) = y/x`
`d/dx(x^{sinx})` = ______
If `("f"(x))/(log (sec x)) "dx"` = log(log sec x) + c, then f(x) = ______.
If xm . yn = (x + y)m+n, prove that `"dy"/"dx" = y/x`
If y = `log ((1 - x^2)/(1 + x^2))`, then `"dy"/"dx"` is equal to ______.
Given f(x) = `log((1 + x)/(1 - x))` and g(x) = `(3x + x^3)/(1 + 3x^2)`, then fog(x) equals
Derivative of log (sec θ + tan θ) with respect to sec θ at θ = `π/4` is ______.
If `log_10 ((x^2 - y^2)/(x^2 + y^2))` = 2, then `dy/dx` is equal to ______.
What is the first step in the standard procedure for \[y=[u(x)]^{v(x)}\]?
What is \[\frac{1}{y}\cdot\frac{dy}{dx}\] for \[y=\sqrt{\frac{(x-3)(x^2+4)}{3x^2+4x+5}}\]?
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?
What condition must be ensured for an expression inside logarithm?
