Advertisements
Advertisements
Question
Find the derivative of the following w. r. t. x. : `(xe^x)/(x+e^x)`
Advertisements
Solution
Let y = `(x"e"^x)/(x + "e"^x)`
Differentiating w.r.t. x, we get
`dy/dx = d/dx((x"e"^x)/(x + "e"^x))`
= `((x + "e"^x)d/dx(x"e"^x) -(x"e"^x)d/dx(x + "e"^x))/(x + "e"^x)^2`
=`((x + "e"^x)[xd/dx("e"^x) + "e"^xd/dx(x)] - x"e"^x(d/dx(x) + d/dx("e"^x)))/(x + "e"^x)^2`
= `((x + "e"^x)[x"e"^x + "e"^x(1)] - x"e"^x(1 + "e"^x))/(x + "e"^x)^2`
=`((x + "e"^x)(x"e"^x + "e"^x) - x"e"^x(1 + "e"^x))/(x + "e"^x)^2`
= `((x + "e"^x)"e"^x(x + 1) - x"e"^x(1 + "e"^x))/(x + "e"^x)^2`
= `("e"^x[(x + "e"^x)(x + 1) - x(1 + "e"^x)])/(x + "e"^x)^2`
APPEARS IN
RELATED QUESTIONS
Find the derivative of the following w. r. t.x. : `(x^2+a^2)/(x^2-a^2)`
Find the derivative of the following w. r. t. x. : `logx/(x^3-5)`
Find the derivative of the following w. r. t.x. : `(3e^x-2)/(3e^x+2)`
Find the derivative of the following function by the first principle: `x sqrtx`
Differentiate the following function w.r.t.x. : `x/(x + 1)`
Differentiate the following function w.r.t.x. : `"e"^x/("e"^x + 1)`
The demand function of a commodity is given as P = 20 + D − D2. Find the rate at which price is changing when demand is 3.
Solve the following example: The total cost of producing x units is given by C = 10e2x, find its marginal cost and average cost when x = 2.
Differentiate the following function w.r.t.x. : x−2
Differentiate the following function w.r.t.x. : `xsqrt x`
Differentiate the followingfunctions.w.r.t.x.: `1/sqrtx`
Find `dy/dx if y=(sqrtx+1)^2`
Find `dy/dx if y = x^3 – 2x^2 + sqrtx + 1`
Find `dy/dx` if y = x2 + 2x – 1
Find `dy/dx` if y = (1 – x) (2 – x)
The relation between price (P) and demand (D) of a cup of Tea is given by D = `12/"P"`. Find the rate at which the demand changes when the price is Rs. 2/-. Interpret the result.
The demand (D) of biscuits at price P is given by D = `64/"P"^3`, find the marginal demand when price is Rs. 4/-.
Differentiate the following w.r.t.x :
y = `x^(4/3) + "e"^x - sinx`
Differentiate the following w.r.t.x :
y = `sqrt(x) + tan x - x^3`
