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 following function with respect to x: `(log x)^x+x^(logx)`
Differentiate the function with respect to x.
(log x)cos x
Differentiate the function with respect to x.
(log x)x + xlog x
Find `bb(dy/dx)` for the given function:
(cos x)y = (cos y)x
If y = `e^(acos^(-1)x)`, −1 ≤ x ≤ 1, show that `(1- x^2) (d^2y)/(dx^2) -x dy/dx - a^2y = 0`.
Evaluate
`int 1/(16 - 9x^2) dx`
Find `"dy"/"dx"` , if `"y" = "x"^("e"^"x")`
Find `"dy"/"dx"` if y = xx + 5x
If `(sin "x")^"y" = "x" + "y", "find" (d"y")/(d"x")`
If log (x + y) = log(xy) + p, where p is a constant, then prove that `"dy"/"dx" = (-y^2)/(x^2)`.
If `log_5((x^4 + y^4)/(x^4 - y^4)) = 2, "show that""dy"/"dx" = (12x^3)/(13y^3)`.
Differentiate 3x w.r.t. logx3.
If y = A cos (log x) + B sin (log x), show that x2y2 + xy1 + y = 0.
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 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.
lf y = `2^(x^(2^(x^(...∞))))`, then x(1 - y logx logy)`dy/dx` = ______
If y = `{f(x)}^{phi(x)}`, then `dy/dx` is ______
If xy = ex-y, then `"dy"/"dx"` at x = 1 is ______.
If y = tan-1 `((1 - cos 3x)/(sin 3x))`, then `"dy"/"dx"` = ______.
`2^(cos^(2_x)`
If xm . yn = (x + y)m+n, prove that `"dy"/"dx" = y/x`
If `f(x) = log [e^x ((3 - x)/(3 + x))^(1/3)]`, then `f^'(1)` is equal to
Given f(x) = `log((1 + x)/(1 - x))` and g(x) = `(3x + x^3)/(1 + 3x^2)`, then fog(x) equals
If y = `(1 + 1/x)^x` then `(2sqrt(y_2(2) + 1/8))/((log 3/2 - 1/3))` is equal to ______.
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`.
Find `dy/dx`, if y = (log x)x.
If \[y=x^x+x^{\frac{1}{x}}\] then \[\frac{\mathrm{d}y}{\mathrm{d}x}\] is equal to
