Math  /  Calculus

QuestionSuppose that x4+y4=82x^{4}+y^{4}=82. (1) Use the method of implicit differentiation to find dydx\frac{d y}{d x}. dydx=\frac{d y}{d x}= \square (2) Find the equation of the tangent line at the point (x,y)=(1,3)(x, y)=(-1,3).
The equation is y=y= \square

Studdy Solution
Find the equation of the tangent line at the point (1,3)(-1, 3). The slope of the tangent line is 127\frac{1}{27}, and it passes through the point (1,3)(-1, 3).
Use the point-slope form of a line:
yy1=m(xx1)y - y_1 = m(x - x_1)
Substitute m=127m = \frac{1}{27}, x1=1x_1 = -1, and y1=3y_1 = 3:
y3=127(x+1)y - 3 = \frac{1}{27}(x + 1)
Simplify to find the equation of the tangent line:
y=127x+127+3y = \frac{1}{27}x + \frac{1}{27} + 3
Combine the constant terms:
y=127x+8227y = \frac{1}{27}x + \frac{82}{27}
The equation of the tangent line is:
y=127x+8227 y = \frac{1}{27}x + \frac{82}{27}

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