Calculus Examples

Find the Tangent Line at the Point 2x^2+y^2=12 , (2,-2)
2x2+y2=122x2+y2=12 , (2,-2)(2,2)
Step 1
Find the first derivative and evaluate at x=2x=2 and y=-2y=2 to find the slope of the tangent line.
Tap for more steps...
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.
Tap for more steps...
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].
Tap for more steps...
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-1nxn1 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].
Tap for more steps...
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.
Tap for more steps...
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.
Tap for more steps...
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.
Tap for more steps...
Step 1.5.2.1
Divide each term in 2yy=-4x by 2y.
2yy2y=-4x2y
Step 1.5.2.2
Simplify the left side.
Tap for more steps...
Step 1.5.2.2.1
Cancel the common factor of 2.
Tap for more steps...
Step 1.5.2.2.1.1
Cancel the common factor.
2yy2y=-4x2y
Step 1.5.2.2.1.2
Rewrite the expression.
yyy=-4x2y
yyy=-4x2y
Step 1.5.2.2.2
Cancel the common factor of y.
Tap for more steps...
Step 1.5.2.2.2.1
Cancel the common factor.
yyy=-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.
Tap for more steps...
Step 1.5.2.3.1
Cancel the common factor of -4 and 2.
Tap for more steps...
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.
Tap for more steps...
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.
Tap for more steps...
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.
-(-12)
Step 1.7.4
Multiply -(-12).
Tap for more steps...
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
Plug the slope and point values into the point-slope formula and solve for y.
Tap for more steps...
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.
Tap for more steps...
Step 2.3.1
Simplify 2(x-2).
Tap for more steps...
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.
Tap for more steps...
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
 [x2  12  π  xdx ]