Use implicit differentiation to find an equation of the tangent line to the graph at the given point.
x + y − 1 = ln(x^5 + y^5), (1, 0)
x + y − 1 = ln(x^5 + y^5)
1 + y' = 1/(x^5+y^5) (5x^4 + 5y^4 y')
y'(1 - 5y^4/(x^5+y^5) = 5x^4/(x^5+y^5) - 1
y' = ((x-5)x^4 - y^5)/(x^5 + (y-5)y^4)
At (1,0),
y' = (-4)/(-1) = 4
So, the line is
y = 4(x-1)
To find the equation of the tangent line to the graph at the given point, we can use implicit differentiation. Here's how to do it:
Step 1: Differentiate both sides of the equation with respect to x, treating y as a function of x:
d/dx (x + y − 1) = d/dx (ln(x^5 + y^5))
Step 2: Simplify the derivatives using the chain rule:
1 + dy/dx - 0 = (1/(x^5 + y^5)) * (d/dx (x^5 + y^5))
Step 3: Simplify the derivative of (x^5 + y^5) using the chain rule:
1 + dy/dx = (1/(x^5 + y^5)) * (5x^4 + 5y^4 * dy/dx)
Step 4: Rearrange the equation to solve for dy/dx:
dy/dx = [(1 + dy/dx) * (x^5 + y^5)] / [5x^4 + 5y^4]
Step 5: Substitute the coordinates of the point (1, 0) into the equation to find the slope of the tangent line:
slope = dy/dx = [(1 + dy/dx(1, 0)) * (1^5 + 0^5)] / [5(1^4) + 5(0^4)]
Step 6: Simplify the expression further:
slope = dy/dx = [1 + dy/dx(1, 0)] / 5
Step 7: Solve for dy/dx(1, 0). Plug in the coordinates (1, 0) into the expression for dy/dx:
dy/dx(1, 0) = [(1 + dy/dx(1, 0)) * (1^5 + 0^5)] / [5(1^4) + 5(0^4)]
Step 8: Solve the equation for dy/dx(1, 0):
dy/dx(1, 0) = 1 / (5 - 1)
dy/dx(1, 0) = 1 / 4
Step 9: Substitute the value of dy/dx(1, 0) into the equation for the slope:
slope = dy/dx = [1 + (1/4)] / 5
slope = 5/20
Step 10: Now we have the slope of the tangent line, and we know that it passes through the point (1, 0). We can use the point-slope form of a line to find the equation of the tangent line:
y - y1 = m(x - x1)
where (x1, y1) is the point (1, 0) and m is the slope 5/20. Plugging in the values, we get:
y - 0 = (5/20)(x - 1)
Simplifying the equation gives:
y = (5/20)(x - 1)
y = (1/4)(x - 1)
Therefore, the equation of the tangent line to the graph at the point (1, 0) is y = (1/4)(x - 1).