What is the tangent of x*sin(pix) ?

The tangent of x*sin(pix) is y=(sin(pia_{0})+pia_{0}cos(pia_{0}))x-pia_{0}^2cos(pia_{0})

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tangent of x*sin(pix)

tangent of

x \cdot sin (π x)

y = (sin (π a_{0}) + π a_{0} cos (π a_{0})) x - π a_{0}^{2} cos (π a_{0})

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

(a_{0}, a_{0} sin (π a_{0}))

Find the slope of

y = x sin (π x) : \frac{d y}{d x} = sin (π x) + π x cos (π x)

Compute the slope of

y = x sin (π x)

x = a_{0} : m = sin (π a_{0}) + π a_{0} cos (π a_{0})

Find the line with slope m=

sin (π a_{0}) + π a_{0} cos (π a_{0})

and passing through

(a_{0}, a_{0} sin (π a_{0})) : y = (sin (π a_{0}) + π a_{0} cos (π a_{0})) x - π a_{0}^{2} cos (π a_{0})

y = (sin (π a_{0}) + π a_{0} cos (π a_{0})) x - π a_{0}^{2} cos (π a_{0})

\int - 3 e^{x} d x

\int (tan (x) + cot (x))^{2} d x

\int - 5 sin (3 x) d x

\frac{\partial}{\partial y} (x^{3} + 3 x y^{2})

\int (\frac{x ^{2}}{2} + sin (x)) d x

What is the tangent of x*sin(pix) ?
The tangent of x*sin(pix) is y=(sin(pia_{0})+pia_{0}cos(pia_{0}))x-pia_{0}^2cos(pia_{0})