Advertisements
Advertisements
प्रश्न
Find `"dy"/"dx"`, if y = `x^tanx + sqrt((x^2 + 1)/2)`
Advertisements
उत्तर
Given that: y = `x^tanx + sqrt((x^2 + 1)/2)`
Let u = `x^tanx` and v = `sqrt((x^2 + 1)/2)`
∴ y = u + v
Differentiating both sides w.r.t. x
`"dy"/"dx" = "du"/"dx" + "dv"/"dx"` .....(i)
Now taking u = `x^tanx`
Taking log on both sides log u = `log(x^tanx)`
log u = tan x . log x
Differentiating both sides w.r.t. x
`1/"u" * "du"/"dx" = "d"/"dx"(tan x * log x)`
⇒ `1/"u" * "du"/"dx" = tan x * "d"/"dx" (log x) + log x * "d"/"dx" (tan x)`
⇒ `1/"u" * "du"/"dx" = tan x * 1/x + log x * sec^2x`
⇒ `"du"/"dx" = "u"[tanx/x + log x * sec^2x]`
∴ `"du"/"dx" = x^tanx [tanx/x + log x sec^2x]`
Taking v = `sqrt((x^2 + 1)/2)`
⇒ v = `1/sqrt(2) sqrt(x^2 + 1)`
Differentiating both sides w.r.t. x
`"dv"/"dx" = 1/sqrt(2) * 1/(2sqrt(x^2 + 1)) * 2x`
= `x/(sqrt(2)sqrt(x^2 + 1))`
Putting the values of `"du"/"dx"` and `"dv"/"dx"` in equation (i)
`"dy"/"dx" = x^tanx [log x sec^2x + tanx/x] + x/(sqrt(2)sqrt(x^2 + 1))`
APPEARS IN
संबंधित प्रश्न
Differentiate 3x w.r.t. log3x
Differentiate the following w.r.t. x:
`e^x/sinx`
Differentiate the following w.r.t. x:
`e^(sin^(-1) x)`
Differentiate the following w.r.t. x:
`e^(x^3)`
Differentiate the following w.r.t. x:
log (cos ex)
Differentiate the following w.r.t. x:
`sqrt(e^(sqrtx))`, x > 0
Differentiate the following w.r.t. x:
log (log x), x > 1
Differentiate the following w.r.t. x:
cos (log x + ex), x > 0
Differentiate the function with respect to x:
(log x)log x, x > 1
Differentiate the function with respect to x:
cos (a cos x + b sin x), for some constant a and b.
If xy - yx = ab, find `(dy)/(dx)`.
The derivative of log10x w.r.t. x is ______.
If yx = ey – x, prove that `"dy"/"dx" = (1 + log y)^2/logy`
If y = `(cos x)^((cos x)^((cosx)....oo)`, show that `"dy"/"dx" = (y^2 tanx)/(y log cos x - 1)`
If `"y" = ("x" + sqrt(1 + "x"^2))^"n", "then" (1 + "x"^2) ("d"^2 "y")/"dx"^2 + "x" ("dy")/("dx")` is ____________.
If `"y = a"^"x", "b"^(2"x" -1), "then" ("d"^2"y")/"dx"^2` is ____________.
If `"y" = (varphi "n x")/"x",` then the value of y'' (e) is ____________.
If `"xy"^2 = "ax"^2 + "bxy" + "y"^2, "then find" "dy"/"dx"`
If `"y = tan"^-1 [("sin x + cos x")/("cos x - sin x")], "then" "dy"/"dx"` is equal to ____________.
For positive values of \[x\], which expression grows faster than \[x^n\] for any fixed positive integer \[n\] when \[x\] is sufficiently large?
What is \[\frac{d}{dx}(e^x)\]?
What is \[\frac{d}{dx}(\log x)\] for \[x>0\]?
What is the range of the exponential function \[y=b^x\]?
Through which point does every graph of \[y=b^x\] pass?
What is the range of the logarithmic function \[y=\log_b x\]?
Through which point does every graph of \[y=\log_b x\] pass?
The graph of \[y=\log_b x\] is a mirror image of which graph reflected perfectly across \[y=x\]?
If \[y=b^x\], which equivalent logarithmic statement is correct?
Which pair of graphs are reflections of each other about \[y=x\]?
Differentiate \[\cos^{-1}(e^x)\] with respect to \[x\].
Which list contains the main log laws?
