Question

find dy/dx if x=at,y=at^2​

Answer 1

Answer:

$\dfrac{dy}{dx} = 2t$

Step-by-step explanation:

Given---->

$x \: = at$

$y \: = a {t}^{2}$

To find ---->

$value \: of \: \dfrac{dy}{dx}$

Concept used---->

1)

$\dfrac{dy}{dx} = \frac{ \dfrac{dy}{dt} }{ \dfrac{dx}{dt} }$

2)

$\dfrac{d}{dx} ( \: {x}^{n} ) = n {x}^{n - 1}$

Solution---->

$x \: = at$

$differentiating \: with \: respect \: to \: x \: we \: get$

$\dfrac{dx}{dt} = \dfrac{d}{dt} ( \: at \: )$

$\: \: = a \: \dfrac{d}{dt} ( \: t \: )$

$\: \: = a \: ( \: 1 \: )$

$\: \: = a$

$now$

$y \: = \: a \: {t}^{2}$

$differentiating \: with \: respect \: to \: x \: we \: get$

$\dfrac{dy}{dt} = \dfrac{d}{dt} ( \: a \: {t}^{2} \: )$

$\dfrac{dy}{dt} \: = a \: \dfrac{d}{dt} \: ( \: {t}^{2} \: )$

$\dfrac{dy}{dt} \: = a \: ( \: 2 \: t \: )$

$\dfrac{dy}{dt} \: = 2 \: a \: t$

$now \:$

$\dfrac{dy}{dx} \: = \frac{ \dfrac{dy}{dt} }{ \dfrac{dx}{dt} }$

$\dfrac{dy}{dx} \: = \dfrac{2 \: a \: t}{a}$

$\dfrac{dy}{dx} \: = 2 \: t$

Additional information ---->

1)

$\dfrac{d}{dx} ( \: {x}^{n} ) = n \: {x}^{n - 1}$

2)

$\dfrac{d}{dx}( {e}^{x} ) \: = {e}^{x}$

3)

$\dfrac{d}{dx} ( \: {a}^{x} \: ) = {a}^{x} log_{e}(a)$