Answer:
Equation of tangent:
[tex]y = xe^{x}[/tex]
At point (1,0):
y = 2.713
Step-by-step explanation:
The equation of tangent line to the function can be calculated by taking the first derivative.
We have,
[tex]y = xe^{x}-e^{x}\\\frac{dy}{dx}=\frac{d}{dx}[ xe^{x} ]-\frac{d}{dx} [e^{x}]\\[/tex]
Applying Product Rule:
d/dx [u.v] = (d/dx u) . (v) + (u) . (d/dx v)
Therefore,
[tex]\frac{dy}{dx}=\frac{d}{dx}(x) . e^{x}+x .\frac{d}{dx}(e^{x})- \frac{d}{dx}(e^{x})\\\\\frac{dy}{dx}=(1)(e^{x})+(x)(e^{x})-e^{x}\\ \frac{dy}{dx}=xe^{x}\\\\[/tex]
The above equation is the equation of tangent line.
The point given is (1,0):
So,
[tex]\frac{dy}{dx} = (1)e^{1}\\ \frac{dy}{dx} = e\\\frac{dy}{dx} = 2.713[/tex]
