Advertisements
Advertisements
Question
Differentiate log (1 + x2) with respect to ax.
Advertisements
Solution
Let u = log (1 + x2) and v = ax
u = log (1 + x2)
Differentiating both sides w.r.t.x, we get
`"du"/"dx" = 1/(1 + "x"^2) * "d"/"dx" (1 + "x"^2)`
`= 1/(1 + "x"^2) * (0 + "2x")`
∴ `"du"/"dx" = "2x"/(1 + "x"^2)`
v = ax
Differentiating both sides w.r.t.x, we get
`"dv"/"dx" = "a"^"x" * log "a"`
∴ `"du"/"dv" = ("du"/"dx")/("dv"/"dx") = ("2x"/(1 + "x"^2))/("a"^"x" * log "a")`
∴ `"du"/"dv" = "2x"/("a"^"x" * log "a" * (1 + "x"^2))`
APPEARS IN
RELATED QUESTIONS
Find `"dy"/"dx"`if, y = `"e"^("x"^"x")`
Find `"dy"/"dx"`if, y = (2x + 5)x
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` = ?
If y = log `("e"^"x"/"x"^2)`, then `"dy"/"dx" = ?`
Fill in the Blank
If 0 = log(xy) + a, then `"dy"/"dx" = (-"y")/square`
The derivative of ax is ax log a.
If u = 5x and v = log x, then `("du")/("dv")` is ______
State whether the following statement is True or False:
If y = log(log x), then `("d"y)/("d"x)` = logx
Find `("d"y)/("d"x)`, if y = [log(log(logx))]2
Find `("d"y)/("d"x)`, if x = `sqrt(1 + "u"^2)`, y = log(1 +u2)
If x = t.logt, y = tt, then show that `("d"y)/("d"x)` = tt
Find `("d"y)/("d"x)`, if y = (log x)x + (x)logx
Find `("d"y)/("d"x)`, if y = xx + (7x – 1)x
Find `("d"y)/("d"x)`, if y = `root(3)(((3x - 1))/((2x + 3)(5 - x)^2)`
If xa .yb = `(x + y)^((a + b))`, then show that `("d"y)/("d"x) = y/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
`int 1/(4x^2 - 1) dx` = ______.
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)`
Find `dy/dx` if, `y = x^(e^x)`
Find `dy/(dx) "if", y = x^(e^(x))`
