Advertisements
Advertisements
प्रश्न
Find `("d"y)/("d"x)`, if y = `x^(x^x)`
Advertisements
उत्तर
y = `x^(x^x)`
Taking logarithm of both sides, we get
log y = `log x^(x^x)`
∴ log y = xx log x
Differentiating both sides w.r.t. x, we get
`"d"/("d"x)(log y) = x^x*"d"/("d"x)(log x) + logx*"d"/("d"x)(x^x)`
∴ `1/y*("d"y)/("d"x) = x^x*1/x + logx*"d"/("d"x)(x^x)` ......(i)
Let u = xx
Taking logarithm of both sides, we get
log u = log xx
∴ log u = x log x
Differentiating both sides w.r.t. x, we get
`"d"/("d"x)(log "u") = x*"d"/("d"x)(log x) + logx*"d"/("d"x)(x)`
∴ `1/"u"*"du"/("d"x) = x*1/x + logx*1`
∴ `1/"u"*"du"/("d"x)` = 1 + log x
∴ `"du"/("d"x)` = u(1 + log x)
∴ `"d"/("d"x)(x^x)` = xx(1 + log x) ......(ii)
Substituting (ii) in (i), we get
`1/y*("d"y)/("d"x) = x^x*1/x + logx*x^x(1 + log x)`
∴ `("d"y)/("d"x) = yx^x[1/x + logx(1 + logx)]`
∴ `("d"y)/("d"x) = x^(x^x)*x^x[1/x + logx(1 + logx)]`
APPEARS IN
संबंधित प्रश्न
Find `"dy"/"dx"`if, y = `"x"^("x"^"2x")`
Find `"dy"/"dx"`if, y = `"x"^("e"^"x")`
Find `"dy"/"dx"`if, y = `(1 + 1/"x")^"x"`
Find `"dy"/"dx"`if, y = `(log "x"^"x") + "x"^(log "x")`
Find `"dy"/"dx"`if, y = `10^("x"^"x") + 10^("x"^10) + 10^(10^"x")`
State whether the following is True or False:
The derivative of `log_ax`, where a is constant is `1/(x.loga)`.
Solve the following:
If y = [log(log(logx))]2, find `"dy"/"dx"`
Find `"dy"/"dx"` if y = `"x"^"x" + ("7x" - 1)^"x"`
If xy = 2x – y, then `("d"y)/("d"x)` = ______
If u = 5x and v = log x, then `("du")/("dv")` is ______
If u = ex and v = loge 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 `(dy)/(dx)`, if xy = yx
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 = `root(3)(((3x - 1))/((2x + 3)(5 - x)^2)`
`int 1/(4x^2 - 1) dx` = ______.
FInd `dy/dx` if,`x=e^(3t), y=e^sqrtt`
Find `dy/dx "if", y = x^(e^x)`
Find `dy/dx` if, y = `x^(e^x)`
Find `dy/dx , if y^x = e^(x+y)`
Find `dy/dx, "if" y=sqrt((2x+3)^5/((3x-1)^3(5x-2)))`
Find `dy/dx,"if" y=x^x+(logx)^x`
Find `dy/dx` if, `y = x^(e^x)`
Find `dy/(dx) "if", y = x^(e^(x))`
