Advertisements
Advertisements
प्रश्न
Differentiate the function with respect to x:
xx + xa + ax + aa, for some fixed a > 0 and x > 0
Advertisements
उत्तर
Let, y = xx + xa + ax + aa
On differentiating with respect to x,
`dy/dx = d/dx (x^x) + d/dx (x^a) + d/dx (a^x) + (a^a) d/dx (1)`
= `d/dx (x^x) + ax^(a - 1) + a^x log a + 0` ...(1)
Let, u = xx
Taking log on both sides,
log u = x log x
On differentiating with respect to x,
`1/u (du)/dx = x d/dx log x + log x d/dx (x)`
= `x * 1/x + log x`
= (1 + log x)
∴ `(du)/dx` = u (1 + log x)
= xx (1 + log x)
i.e., `d/dx (x^x) = (du)/dx`
= xx (1 + log x)
Putting the value of `d/dx (x^x)` in equation (1),
`dy/dx` = xx (1 + log x) + axa − 1 + ax log a
APPEARS IN
संबंधित प्रश्न
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
if xx+xy+yx=ab, then find `dy/dx`.
Differentiate the function with respect to x.
cos x . cos 2x . cos 3x
Differentiate the function with respect to x.
(log x)cos x
Differentiate the function with respect to x.
xx − 2sin x
Differentiate the function with respect to x.
(log x)x + xlog x
Find the derivative of the function given by f(x) = (1 + x) (1 + x2) (1 + x4) (1 + x8) and hence find f′(1).
Differentiate (x2 – 5x + 8) (x3 + 7x + 9) in three ways mentioned below:
- By using the product rule.
- By expanding the product to obtain a single polynomial.
- By logarithmic differentiation.
Do they all give the same answer?
If ey ( x +1) = 1, then show that `(d^2 y)/(dx^2) = ((dy)/(dx))^2 .`
Evaluate
`int 1/(16 - 9x^2) dx`
Find `(d^2y)/(dx^2)` , if y = log x
Solve the following differential equation: (3xy + y2) dx + (x2 + xy) dy = 0
`"If" y = sqrt(logx + sqrt(log x + sqrt(log x + ... ∞))), "then show that" dy/dx = (1)/(x(2y - 1).`
If x = 2cos4(t + 3), y = 3sin4(t + 3), show that `"dy"/"dx" = -sqrt((3y)/(2x)`.
If x = log(1 + t2), y = t – tan–1t,show that `"dy"/"dx" = sqrt(e^x - 1)/(2)`.
If x = `(2bt)/(1 + t^2), y = a((1 - t^2)/(1 + t^2)), "show that" "dx"/"dy" = -(b^2y)/(a^2x)`.
Find the second order derivatives of the following : x3.logx
Find the second order derivatives of the following : log(logx)
Choose the correct option from the given alternatives :
If xy = yx, then `"dy"/"dx"` = ..........
If y = 5x. x5. xx. 55 , find `("d"y)/("d"x)`
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 = tan-1 `((1 - cos 3x)/(sin 3x))`, then `"dy"/"dx"` = ______.
`"d"/"dx" [(cos x)^(log x)]` = ______.
Derivative of `log_6`x with respect 6x to is ______
If xm . yn = (x + y)m+n, prove that `"dy"/"dx" = y/x`
If y = `log ((1 - x^2)/(1 + x^2))`, then `"dy"/"dx"` is equal to ______.
`lim_("x" -> 0)(1 - "cos x")/"x"^2` is equal to ____________.
If `"y" = "e"^(1/2log (1 + "tan"^2"x")), "then" "dy"/"dx"` is equal to ____________.
If y `= "e"^(3"x" + 7), "then the value" |("dy")/("dx")|_("x" = 0)` is ____________.
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 ______.
The derivative of log x with respect to `1/x` is ______.
Find `dy/dx`, if y = (log x)x.
Find the derivative of `y = log x + 1/x` with respect to x.
