What is the tangent of y=x+e^{2x} ?

The tangent of y=x+e^{2x} is y=(1+e^{2a_{0}}*2)x-2a_{0}e^{2a_{0}}+e^{2a_{0}}

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tangent of y=x+e^{2x}

tangent of

y = x + e^{2 x}

y = (1 + e^{2 a_{0}} \cdot 2) x - 2 a_{0} e^{2 a_{0}} + e^{2 a_{0}}

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

(a_{0}, a_{0} + e^{2 a_{0}})

Find the slope of

y = x + e^{2 x} : \frac{d y}{d x} = 1 + e^{2 x} \cdot 2

Compute the slope of

y = x + e^{2 x}

x = a_{0} : m = 1 + e^{2 a_{0}} \cdot 2

Find the line with slope m=

1 + e^{2 a_{0}} 2

and passing through

(a_{0}, a_{0} + e^{2 a_{0}}) : y = (1 + e^{2 a_{0}} \cdot 2) x - 2 a_{0} e^{2 a_{0}} + e^{2 a_{0}}

y = (1 + e^{2 a_{0}} \cdot 2) x - 2 a_{0} e^{2 a_{0}} + e^{2 a_{0}}

x^{3}, x, - 1, 1

\int 6 x^{5} sin (x^{6}) d x

\frac{x ^{4} + 2 x ^{2}}{x ^{2}}

\int e^{x} (x^{2} - 5 x) d x

\frac{d y}{d x} = \frac{x ^{3}}{y ^{2}}

What is the tangent of y=x+e^{2x} ?
The tangent of y=x+e^{2x} is y=(1+e^{2a_{0}}*2)x-2a_{0}e^{2a_{0}}+e^{2a_{0}}