Find an equation of the tangent line to the curve...

Question

1)Find an equation of the tangent line to the curve e^(xy) + x^2 = 2 − sin(xy) at (1,0).

2)Find y ′′ at the point (3,3) if x^3 + y^3 = 54 .

Answer 1

Step 1 :

1) The equation of the curve is and the point .

Slope of the tangent line is  .

Use the chain rule to find the derivative.

Chain rule of differentiation is .

Differentiate with respect to on each side.

Slope of the tangent line at .

Slope of the tangent line .

Point-slope form of line equation .

Substitute slope and the point .

The equation of a tangent line

Step 2 :

2) The equation of the curve is and the point .

Differentiate with respect to on each side.

Consider .

Differentiate with respect to on each side.

Substitute in the above equation.

Find at the point .

Solution : 

1) The equation of a tangent line

2)