To determine

To find: An equation of the tangent line to the curve y at the point (π,π).

Answer

The equation of the tangent line at (π,π) is y=2x−π_.

Explanation

Given:

The function is y=x+tanx at the point (π,π).

Formula used:

Sum Rule:

If f(x) and g(x) are both differentiable function, then the sum rule is,

ddx[f(x)+g(x)]=ddx[f(x)]+ddx[g(x)] (1)

The slope of the tangent line y is, dydx=m at x1 (2)

The equation of the tangent line is, (y−y1)=m(x−x1) at the point (x1,y1) (3)

Calculation:

Find derivative of the function y=x+tanx using Sum Rule as shown in equation (1).

ddx[x+tanx]=ddx[x]+ddx[tanx]=[1]+[sec2x]=1+sec2x

Substitute the value 1+sec2x for dydx in equation (2),

m=1+sec2x

Substitute π for x,

m=1+sec2π=1+1=2

Then the slope of the tangent line is 2.

The equation of the tangent line at the point (π,π) is calculated using equation (3).

Substitute π for y1, π for x1, and 2 for m in equation (3),

(y−π)=2(x−π)y−π=2x−2πy=2x−2π+πy=2x−π

Thus, the equation of the tangent line at (π,π) is y=2x−π_.