Advertisements
Advertisements
प्रश्न
Fill in the blank.
If y = y = [log (x)]2 then `("d"^2"y")/"dx"^2 =` _____.
उत्तर
If y = y = [log (x)]2 then `("d"^2"y")/"dx"^2 =(2(1 − log x))/x^2 `.
Explanation:
y = (log x)2
On differentiating w.r.t. x, we get,
`dy/dx = 2 log x d/dx (log x)`
`dy/dx = 2 log x. 1/x`
`dy/dx = (2log x)/x`
Again differentiating w.r to x, we get,
`(d^2y)/(dx^2) = 2 d/dx ((log x)/x)`
`(d^2y)/(dx^2) = 2 ((x d/dx (log x) − log x d/dx x)/x^2)`
`(d^2y)/(dx^2) = 2 ((x × 1/x − log x × 1)/x^2)`
`(d^2y)/(dx^2) = (2(1 − log x))/x^2`
APPEARS IN
संबंधित प्रश्न
Find `"dy"/"dx"`if, y = `"e"^("x"^"x")`
Find `"dy"/"dx"`if, y = `(1 + 1/"x")^"x"`
Find `"dy"/"dx"`if, y = `root(3)(("3x" - 1)/(("2x + 3")(5 - "x")^2))`
Find `"dy"/"dx"`if, y = `(log "x"^"x") + "x"^(log "x")`
Find `dy/dx`if, y = `(x)^x + (a^x)`.
If y = elogx then `dy/dx` = ?
State whether the following is True or False:
The derivative of `log_ax`, where a is constant is `1/(x.loga)`.
The derivative of ax is ax log a.
If y = `"a"^((1 + log"x"))`, then `("d"y)/("d"x)` is ______
State whether the following statement is True or False:
If y = 4x, then `("d"y)/("d"x)` = 4x
Find `("d"y)/("d"x)`, if xy = log(xy)
If x = t.logt, y = tt, then show that `("d"y)/("d"x)` = tt
Find `("d"y)/("d"x)`, if y = xx + (7x – 1)x
Find `("d"y)/("d"x)`, if y = x(x) + 20(x)
Solution: Let y = x(x) + 20(x)
Let u = `x^square` and v = `square^x`
∴ y = u + v
Diff. w.r.to x, we get
`("d"y)/("d"x) = square/("d"x) + "dv"/square` .....(i)
Now, u = xx
Taking log on both sides, we get
log u = x × log x
Diff. w.r.to x,
`1/"u"*"du"/("d"x) = x xx 1/square + log x xx square`
∴ `"du"/("d"x)` = u(1 + log x)
∴ `"du"/("d"x) = x^x (1 + square)` .....(ii)
Now, v = 20x
Diff.w.r.to x, we get
`"dv"/("d"x") = 20^square*log(20)` .....(iii)
Substituting equations (ii) and (iii) in equation (i), we get
`("d"y)/("d"x)` = xx(1 + log x) + 20x.log(20)
Solve the following differential equations:
x2ydx – (x3 – y3)dy = 0
Find`dy/dx if, y = x^(e^x)`
Find `dy/dx "if", y = x^(e^x)`
Find `dy/dx if, y = x^(e^x)`
Find `dy / dx` if, `y = x^(e^x)`
Find `dy/dx` if, `y = x^(e^x)`
