**Use implicit differentiation to find dy/dx.**
**Next, substitute the given point to find the slope of the tangent line.**
**Then find the equation of the tangent line by plugging the slope and the given point and write it in the slope-intercept form.**

_____________________________________________________________________________________

**Step 1: Use implicit differentiation to find dy/dx.**

Given curve equation; y+3x^{2}y^{5}= 4x+3

Now differentiate implicitly i.e, (dy/dx)+ 6xy^5+15x^2y^4(dy/dx)=4+0 (Note: Product rule must be applied to the second term in the equation)

Next, make (dy/dx) the subject of the formula shown below:

(dy/dx)[1+15x^2y^4]+6xy^5=4

(dy/dx)[1+15x^2y^4]=4-6xy^5 (Divide both sides by [1+15x^2y^4])

(dy/dx)= (4-6xy^5)/[1+15x^2y^4]

**Step 2:** **Next, substitute the given point to find the slope of the tangent line.**

Substitute (0,3) to (dy/dx) equation.

(dy/dx)= (4-6(0)(3)^5)/[1+15(0)^2(3)^4]

= 4

so m=4 aka slope.

**Step 3: Then find the equation of the tangent line by plugging the slope and the given point and write it in the slope-intercept form.**

First find b. Substitute (0,3) and m=4 to y=mx+b

3=4(0)+b. So b=3

Therefore the equation of the tangent line is y=4x+3