Advertisements
Advertisements
Question
Find the derivative of the following function by the first principle: `x sqrtx`
Advertisements
Solution
Let f(x) = `xsqrt x = x^(3/2)`
∴ f(x + h) = `(x + "h")^(3/2)`
By first principle, we get
f ′(x) =` lim_("h" → 0) ("f"(x + "h") - "f"(x))/"h"`
=`lim_("h" → 0)((x + "h")^(3/2) - x^(3/2))/"h"`
=`lim_("h" → 0) ([(x + "h")^(3/2) - x^(3/2)][(x+h)^(3/2)+x^(3/2)])/(h[(x+h)^(3/2)+x^(3/2)])`
=`lim_("h" → 0) ((x + "h")^3 - x^3)/("h"[(x + "h")^(3/2) + x^(3/2)])`
=`lim_("h" → 0) (x^3 + 3x^2"h" + 3x"h"^2 + "h"^3 -x^3)/("h"[(x + "h")^(3/2)+x^(3/2))`
= `lim_("h" → 0) ("h"(3x^2 + 3x"h" + "h"^2))/("h"[(x + "h")^(3/2) + x^(3/2)]]`
= `lim_("h" → 0)("h"(3x^2 + 3x"h" + "h"^2))/("h"[(x + "h")^(3/2) + x^(3/2)]`
= `lim_("h" → 0)(3x^2 + 3x"h" + "h"^2)/((x + "h")^(3/2) + x^(3/2))` ...[∵ h → 0, ∴ h ≠ 0]
= `(3x^2 + 3x xx0 + 0^2)/((x + 0)^(3/2) + x^(3/2))`
= `(3x^2)/(2x^(3/2)`
=`3/2x^(1/2)`
= `3/2sqrtx`
APPEARS IN
RELATED QUESTIONS
Find the derivative of the following w. r. t. x. : `logx/(x^3-5)`
Find the derivative of the following function by the first principle: 3x2 + 4
Differentiate the following function w.r.t.x. : `((2"e"^x - 1))/((2"e"^x + 1))`
Differentiate the following function w.r.t.x. : `((x+1)(x-1))/(("e"^x+1))`
If for a commodity; the price-demand relation is given as D =`("P"+ 5)/("P" - 1)`. Find the marginal demand when price is 2.
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: If the total cost function is given by; C = 5x3 + 2x2 + 7; find the average cost and the marginal cost when x = 4.
Solve the following example: The total cost of ‘t’ toy cars is given by C=5(2t)+17. Find the marginal cost and average cost at t = 3.
Solve the following example: If for a commodity; the demand function is given by, D = `sqrt(75 − 3"P")`. Find the marginal demand function when P = 5.
Differentiate the following function w.r.t.x. : x−2
Differentiate the following function w.r.t.x. : `xsqrt x`
Find `dy/dx if y = (sqrtx + 1/sqrtx)^2`
Find `dy/dx if y = ((logx+1))/x`
Find `dy/dx`if y = x log x (x2 + 1)
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.
If the total cost function is given by C = 5x3 + 2x2 + 1; Find the average cost and the marginal cost when x = 4.
Differentiate the following w.r.t.x :
y = `sqrt(x) + tan x - x^3`
Differentiate the following w.r.t.x :
y = `log x - "cosec" x + 5^x - 3/(x^(3/2))`
Select the correct answer from the given alternative:
If y = `(3x + 5)/(4x + 5)`, then `("d"y)/("d"x)` =
