Answer:
If [tex]e^{x\cdot y}-y^{2} = e-4[/tex], then at [tex]x = \frac{1}{2}[/tex] and [tex]y = 2[/tex], [tex]\frac{dy}{dx} \approx 2.059[/tex].
Step-by-step explanation:
Let [tex]e^{x\cdot y}-y^{2} = e-4[/tex]. At first we determine the first derivative of the function by implicit differentiation:
[tex]e^{x\cdot y}\cdot \left(y+x\cdot \frac{dy}{dx} \right)-2\cdot y \cdot \frac{dy}{dx} = 0[/tex]
[tex]y\cdot e^{x\cdot y}+x\cdot e^{x\cdot y}\cdot \frac{dy}{dx} -2\cdot y\cdot \frac{dy}{dx} = 0[/tex]
[tex]y\cdot e^{x\cdot y}= (2\cdot y -x\cdot e^{x\cdot y})\cdot \frac{dy}{dx}[/tex]
[tex]\frac{dy}{dx} = \frac{y\cdot e^{x\cdot y}}{2\cdot y - x\cdot e^{x\cdot y}}[/tex]
If we know that [tex]x = \frac{1}{2}[/tex] and [tex]y = 2[/tex], then the computed value of the function is:
[tex]\frac{dy}{dx} = \frac{(2)\cdot e^{\left(\frac{1}{2} \right)\cdot (2)}}{2\cdot (2)-\left(\frac{1}{2} \right)\cdot e^{\left(\frac{1}{2} \right)\cdot (2)}}[/tex]
[tex]\frac{dy}{dx} \approx 2.059[/tex]
If [tex]e^{x\cdot y}-y^{2} = e-4[/tex], then at [tex]x = \frac{1}{2}[/tex] and [tex]y = 2[/tex], [tex]\frac{dy}{dx} \approx 2.059[/tex].
