Advertisements
Advertisements
Question
Find `("d"^2"y")/"dx"^2`, if y = `"x"^2 * "e"^"x"`
Advertisements
Solution
y = `"x"^2 * "e"^"x"`
Differentiating both sides w.r.t. t, we get
`"dy"/"dx" = "x"^2 * "d"/"dx" ("e"^"x") * "d"/"dx" ("x"^2)`
`= "x"^2 * "e"^"x" + "e"^"x" ("2x")`
`"dy"/"dx" = ("x"^2 + 2"x") * "e"^"x"`
Again, differentiating both sides w.r.t. x, we get
`("d"^2"y")/"dx"^2 = ("x"^2 + 2"x") * "d"/"dx" ("e"^"x") + "e"^"x" * "d"/"dx" ("x"^2 + 2"x")`
`= ("x"^2 + 2"x") * "e"^"x" + "e"^"x"`(2x + 2)
`= "e"^"x"("x"^2 + 2"x" + 2"x" + 2)`
∴ `("d"^2"y")/"dx"^2 = "e"^"x"("x"^2 + 4"x" + 2)`
APPEARS IN
RELATED QUESTIONS
If x = a cos θ + b sin θ, y = a sin θ − b cos θ, show that `y^2 (d^2y)/(dx^2)-xdy/dx+y=0`
Find the second order derivative of the function.
x2 + 3x + 2
Find the second order derivative of the function.
x20
Find the second order derivative of the function.
log x
Find the second order derivative of the function.
ex sin 5x
If y = cos–1 x, find `(d^2y)/dx^2` in terms of y alone.
If y = 3 cos (log x) + 4 sin (log x), show that x2y2 + xy1 + y = 0.
If y = 500e7x + 600e–7x, show that `(d^2y)/(dx^2)` = 49y.
If x7 . y9 = (x + y)16 then show that `"dy"/"dx" = "y"/"x"`
Find `("d"^2"y")/"dx"^2`, if y = `"x"^5`
Find `("d"^2"y")/"dx"^2`, if y = `"e"^"log x"`
If x2 + 6xy + y2 = 10, then show that `("d"^2y)/("d"x^2) = 80/(3x + y)^3`
If ax2 + 2hxy + by2 = 0, then show that `("d"^2"y")/"dx"^2` = 0
sec(x + y) = xy
tan–1(x2 + y2) = a
The derivative of cos–1(2x2 – 1) w.r.t. cos–1x is ______.
If y = 5 cos x – 3 sin x, then `("d"^2"y")/("dx"^2)` is equal to:
Derivative of cot x° with respect to x is ____________.
Read the following passage and answer the questions given below:
|
The relation between the height of the plant ('y' in cm) with respect to its exposure to the sunlight is governed by the following equation y = `4x - 1/2 x^2`, where 'x' is the number of days exposed to the sunlight, for x ≤ 3.
|
- Find the rate of growth of the plant with respect to the number of days exposed to the sunlight.
- Does the rate of growth of the plant increase or decrease in the first three days? What will be the height of the plant after 2 days?
`"Find" (d^2y)/(dx^2) "if" y=e^((2x+1))`
Find `(d^2y)/dx^2 if, y = e^((2x + 1))`
Find `(d^2y)/dx^2` if, `y = e^((2x + 1))`
Find `(d^2y)/dx^2` if, `y = e^((2x + 1))`
Find `(d^2y)/(dx^2)` if, y = `e^((2x+1))`
If y = 3 cos(log x) + 4 sin(log x), show that `x^2 (d^2y)/(dx^2) + x dy/dx + y = 0`
Find `(d^2y)/dx^2, "if" y = e^((2x+1))`
Find `(d^2y)/(dx^2) "if", y = e^((2x + 1))`
