Use implicit differentiation to find an equation

of the tangent line to the curve
y^2 = x^3 (26 − x)
at the point (1, 5).

solution:

y^2=x^3(26-x) @ (1,5)
2ydy=3x^2dx(26-x)+(-dx)x^3
[2ydy=3x^2dx(26-x)+(-dx)x^3]1/dx
2ydy/dx=3x^2(26-x)-x^3
dy/dx=[3x^2(26-x)-x^3]/2y
substitute (1,5)
dy/dx=[3(1)^2(26-1)-(1)^3]/2(5)
using calculator:
dy/dx=74/10 or 37/5
then dy/dx=m or slope: we have
m=37/5
equation of line is:
y-y1=m(x-x1) where y1 & x1 is y sub 1
substitute m=37/5 (1,5) to x1 & y1;we have
y-1=37/5(x-5)
y-1=37/5 x-37/25
y=37/5 x-37/25+1
y=37/5 x-12/25
[y=37/5 x-12/25](25)
25y=185x-12

185x-25y=12 (ans)

To find the equation of the tangent line to the curve at a specific point using implicit differentiation, follow these steps:

Step 1: Differentiate both sides of the given equation with respect to x, treating y as a function of x.

Differentiating y^2 = x^3(26 − x) with respect to x, we get:
2y(dy/dx) = 3x^2(26 - x) - x^3

Step 2: Solve the resulting equation for dy/dx, which represents the slope of the tangent line.

Let's rearrange the equation:
2y(dy/dx) = 3x^2(26 - x) - x^3
2y(dy/dx) = 78x^2 - 3x^3 - x^3
2y(dy/dx) = 78x^2 - 4x^3

Divide both sides by 2y:
dy/dx = (78x^2 - 4x^3) / (2y)
dy/dx = (39x^2 - 2x^3) / y

Step 3: Substitute the coordinates of the given point (1, 5) into the equation obtained in Step 2 to find the slope of the tangent line at that point.

Substituting x = 1 and y = 5 into dy/dx = (39x^2 - 2x^3) / y, we get:
dy/dx = (39(1)^2 - 2(1)^3) / 5
dy/dx = (39 - 2) / 5
dy/dx = 37 / 5
dy/dx = 7.4

So, the slope of the tangent line at the point (1, 5) is 7.4.

Step 4: Use the point-slope form of the equation of a line to find the equation of the tangent line.

The equation of a line in point-slope form is given by:
y - y₁ = m(x - x₁)

Substituting the values of the slope (m = 7.4) and the given point (x₁ = 1, y₁ = 5) into the equation, we get:
y - 5 = 7.4(x - 1)

Simplifying further:
y - 5 = 7.4x - 7.4
y = 7.4x - 2.4

Therefore, the equation of the tangent line to the given curve at the point (1, 5) is y = 7.4x - 2.4.