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Calculus Examples
2x2+y2=122x2+y2=12 , (2,-2)(2,−2)
Step 1
Step 1.1
Differentiate both sides of the equation.
ddx(2x2+y2)=ddx(12)ddx(2x2+y2)=ddx(12)
Step 1.2
Differentiate the left side of the equation.
Step 1.2.1
By the Sum Rule, the derivative of 2x2+y22x2+y2 with respect to xx is ddx[2x2]+ddx[y2]ddx[2x2]+ddx[y2].
ddx[2x2]+ddx[y2]ddx[2x2]+ddx[y2]
Step 1.2.2
Evaluate ddx[2x2]ddx[2x2].
Step 1.2.2.1
Since 22 is constant with respect to xx, the derivative of 2x22x2 with respect to xx is 2ddx[x2]2ddx[x2].
2ddx[x2]+ddx[y2]2ddx[x2]+ddx[y2]
Step 1.2.2.2
Differentiate using the Power Rule which states that ddx[xn]ddx[xn] is nxn-1nxn−1 where n=2n=2.
2(2x)+ddx[y2]2(2x)+ddx[y2]
Step 1.2.2.3
Multiply 22 by 22.
4x+ddx[y2]4x+ddx[y2]
4x+ddx[y2]4x+ddx[y2]
Step 1.2.3
Evaluate ddx[y2]ddx[y2].
Step 1.2.3.1
Differentiate using the chain rule, which states that ddx[f(g(x))]ddx[f(g(x))] is f′(g(x))g′(x) where f(x)=x2 and g(x)=y.
Step 1.2.3.1.1
To apply the Chain Rule, set u as y.
4x+ddu[u2]ddx[y]
Step 1.2.3.1.2
Differentiate using the Power Rule which states that ddu[un] is nun-1 where n=2.
4x+2uddx[y]
Step 1.2.3.1.3
Replace all occurrences of u with y.
4x+2yddx[y]
4x+2yddx[y]
Step 1.2.3.2
Rewrite ddx[y] as y′.
4x+2yy′
4x+2yy′
Step 1.2.4
Reorder terms.
2yy′+4x
2yy′+4x
Step 1.3
Since 12 is constant with respect to x, the derivative of 12 with respect to x is 0.
0
Step 1.4
Reform the equation by setting the left side equal to the right side.
2yy′+4x=0
Step 1.5
Solve for y′.
Step 1.5.1
Subtract 4x from both sides of the equation.
2yy′=-4x
Step 1.5.2
Divide each term in 2yy′=-4x by 2y and simplify.
Step 1.5.2.1
Divide each term in 2yy′=-4x by 2y.
2yy′2y=-4x2y
Step 1.5.2.2
Simplify the left side.
Step 1.5.2.2.1
Cancel the common factor of 2.
Step 1.5.2.2.1.1
Cancel the common factor.
2yy′2y=-4x2y
Step 1.5.2.2.1.2
Rewrite the expression.
yy′y=-4x2y
yy′y=-4x2y
Step 1.5.2.2.2
Cancel the common factor of y.
Step 1.5.2.2.2.1
Cancel the common factor.
yy′y=-4x2y
Step 1.5.2.2.2.2
Divide y′ by 1.
y′=-4x2y
y′=-4x2y
y′=-4x2y
Step 1.5.2.3
Simplify the right side.
Step 1.5.2.3.1
Cancel the common factor of -4 and 2.
Step 1.5.2.3.1.1
Factor 2 out of -4x.
y′=2(-2x)2y
Step 1.5.2.3.1.2
Cancel the common factors.
Step 1.5.2.3.1.2.1
Factor 2 out of 2y.
y′=2(-2x)2(y)
Step 1.5.2.3.1.2.2
Cancel the common factor.
y′=2(-2x)2y
Step 1.5.2.3.1.2.3
Rewrite the expression.
y′=-2xy
y′=-2xy
y′=-2xy
Step 1.5.2.3.2
Move the negative in front of the fraction.
y′=-2xy
y′=-2xy
y′=-2xy
y′=-2xy
Step 1.6
Replace y′ with dydx.
dydx=-2xy
Step 1.7
Evaluate at x=2 and y=-2.
Step 1.7.1
Replace the variable x with 2 in the expression.
-2(2)y
Step 1.7.2
Replace the variable y with -2 in the expression.
-2(2)-2
Step 1.7.3
Move the negative one from the denominator of 2-1.
-(-1⋅2)
Step 1.7.4
Multiply -(-1⋅2).
Step 1.7.4.1
Multiply -1 by 2.
--2
Step 1.7.4.2
Multiply -1 by -2.
2
2
2
2
Step 2
Step 2.1
Use the slope 2 and a given point (2,-2) to substitute for x1 and y1 in the point-slope form y-y1=m(x-x1), which is derived from the slope equation m=y2-y1x2-x1.
y-(-2)=2⋅(x-(2))
Step 2.2
Simplify the equation and keep it in point-slope form.
y+2=2⋅(x-2)
Step 2.3
Solve for y.
Step 2.3.1
Simplify 2⋅(x-2).
Step 2.3.1.1
Rewrite.
y+2=0+0+2⋅(x-2)
Step 2.3.1.2
Simplify by adding zeros.
y+2=2⋅(x-2)
Step 2.3.1.3
Apply the distributive property.
y+2=2x+2⋅-2
Step 2.3.1.4
Multiply 2 by -2.
y+2=2x-4
y+2=2x-4
Step 2.3.2
Move all terms not containing y to the right side of the equation.
Step 2.3.2.1
Subtract 2 from both sides of the equation.
y=2x-4-2
Step 2.3.2.2
Subtract 2 from -4.
y=2x-6
y=2x-6
y=2x-6
y=2x-6
Step 3