Question

find y' if y = x+1/✓x, x=1/4​

Answer 1

Given :

• y = x+1/✓x
• x = 1/4

To find :

• y' or dy/dx

Solution :

$\implies \: y = x + \dfrac{1}{ \sqrt{x} }$

Differentiating both sides.

$\implies \: \dfrac{dy}{ dx } =\dfrac{d(x + \dfrac{1}{ \sqrt{x} } )}{ dx }$

$\implies \: \dfrac{dy}{ dx } =\dfrac{d(x )}{ dx } + \dfrac{d( \dfrac{1}{ \sqrt{x} } )}{ dx }$

$\implies \: \dfrac{dy}{ dx } =1 + \dfrac{d( {x}^{ - \frac{1}{2} } )}{ dx }$

$\implies \: \dfrac{dy}{ dx } =1 - \dfrac{1}{2} ( {x}^{ - \frac{1}{2} - 1 })$

$\implies \: \dfrac{dy}{ dx } =1 - \dfrac{1}{2} {x}^{ \frac{ - 3}{2} }$

Now put x = 1/4

$\implies \: \dfrac{dy}{ dx } =1 - \dfrac{1}{2} {( \dfrac{1}{4} )}^{ \frac{ - 3}{2} }$

$\implies \: \dfrac{dy}{ dx } =1 - \dfrac{1}{2} {( \dfrac{1}{ {2}^{2} } )}^{ \frac{ - 3}{2} }$

$\implies \: \dfrac{dy}{ dx } =1 - \dfrac{1}{2} {( \dfrac{1}{ {2} } )}^{ \frac{ - 3}{2} \times 2}$

$\implies \: \dfrac{dy}{ dx } =1 - \dfrac{1}{2} {( \dfrac{1}{ {2} } )}^{ - 3}$

$\implies \: \dfrac{dy}{ dx } =1 - \dfrac{1}{2} {( { {2} } )}^{ 3}$

$\implies \: \dfrac{dy}{ dx } =1 - {( { {2} } )}^{ 2}$

$\implies \: \dfrac{dy}{ dx } =1 - 4$

$\implies \: \dfrac{dy}{ dx } = - 3$

Answer :

y' or dy/dx = -3

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Learn more :

d(x^n)/dx = n x^(n-1)

d(log x)/dx = 1/x

d(e^x)/dx = e^x

d(sinx)/dx = cosx

d(cos x)/dx = -sin x

d(cosec x)/dx = -cot x cosec x

d(tan x)/dx = sec²x

d(sec x)/dx = secx tanx

d(cot x)/dx = - cosec² x