Advertisements
Advertisements
Question
Find `"dy"/"dx"` , if `"y" = "x"^("e"^"x")`
Solution
`"y" = "x"^("e"^"x")`
Taking logarithm of both the sides,
log y = ex log x
Differentiating both sides w.r.t. x,
`1/"y" "dy"/"dx" = "e"^"x" . "d"/"dx"(log"x") + "log x" ."d"/"dx" ("e"^"x")`
`= "e"^"x". 1/"x" + "log x". "e"^"x"`
`= "e"^"x"(1/"x" + "log x")`
`therefore "dy"/"dx" = "y".["e"^"x" (1/"x" + "log x")]`
`= "x"^("e"^"x") . "e"^"x" (1/"x" + "log x")`
RELATED QUESTIONS
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
Differentiate (x2 – 5x + 8) (x3 + 7x + 9) in three ways mentioned below:
- by using product rule
- by expanding the product to obtain a single polynomial.
- by logarithmic differentiation.
Do they all give the same answer?
Differentiate w.r.t. x the function:
xx + xa + ax + aa, for some fixed a > 0 and x > 0
If x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`
if `x^m y^n = (x + y)^(m + n)`, prove that `(d^2y)/(dx^2)= 0`
Evaluate
`int 1/(16 - 9x^2) dx`
Find `(d^2y)/(dx^2)` , if y = log x
If `"x"^(5/3) . "y"^(2/3) = ("x + y")^(7/3)` , the show that `"dy"/"dx" = "y"/"x"`
Solve the following differential equation: (3xy + y2) dx + (x2 + xy) dy = 0
If `(sin "x")^"y" = "x" + "y", "find" (d"y")/(d"x")`
If `log_10((x^3 - y^3)/(x^3 + y^3)) = 2, "show that" "dy"/"dx" = -(99x^2)/(101y^2)`
If y = `x^(x^(x^(.^(.^.∞))`, then show that `"dy"/"dx" = y^2/(x(1 - logy).`.
If x = esin3t, y = ecos3t, then show that `dy/dx = -(ylogx)/(xlogy)`.
Find the second order derivatives of the following : x3.logx
Find the nth derivative of the following : log (2x + 3)
If y = log [cos(x5)] then find `("d"y)/("d"x)`
If y = 5x. x5. xx. 55 , find `("d"y)/("d"x)`
If y = `(sin x)^sin x` , then `"dy"/"dx"` = ?
The rate at which the metal cools in moving air is proportional to the difference of temperatures between the metal and air. If the air temperature is 290 K and the metal temperature drops from 370 K to 330 K in 1 O min, then the time required to drop the temperature upto 295 K.
If y = `{f(x)}^{phi(x)}`, then `dy/dx` is ______
If xy = ex-y, then `"dy"/"dx"` at x = 1 is ______.
`d/dx(x^{sinx})` = ______
If y = `("e"^"2x" sin x)/(x cos x), "then" "dy"/"dx" = ?`
Derivative of `log_6`x with respect 6x to is ______
If xm . yn = (x + y)m+n, prove that `"dy"/"dx" = y/x`
If `f(x) = log [e^x ((3 - x)/(3 + x))^(1/3)]`, then `f^'(1)` is equal to
Given f(x) = `log((1 + x)/(1 - x))` and g(x) = `(3x + x^3)/(1 + 3x^2)`, then fog(x) equals
If y = `(1 + 1/x)^x` then `(2sqrt(y_2(2) + 1/8))/((log 3/2 - 1/3))` is equal to ______.
