Worked examples
The question
Question: find dy/dx for x² + y² = 25.
x2 + y2 = 25
y is not written alone, but y still depends on x.
So for y we use the chain rule.
y squared gives 2y, times dy/dx.
y depends on x: y2 → 2y × dydx
1 The derivative of each side
Now each term.
x terms as usual, y terms times dy/dx, a number alone gives 0.
2x + 2y dydx = 0
Careful!
y depends on x.
So we must multiply by dy/dx.
Careful: not y2 → 2y ✗
2 Get dy/dx alone
Keep the dy/dx terms on the left.
Move the others to the right: they change sign.
2y dydx = −2x
To remove times 2y, we divide both sides by it.
dydx = −2x2y
Every number on the top and the bottom divides by 2.
top and bottom ÷ 2
So dy/dx = −x over y.
dydx = −xy
3 Check
Let's check at a point on the curve: (3, 4).
Our dy/dx gives −0.75 there.
at (3, 4): dy/dx = −0.75
Now a very close point on the curve.
At x = 3.001, a calculator gives y ≈ 3.99925.
x = 3.001: y ≈ 3.99925
The slope between the two points: down 0.00075, across 0.001.
About −0.75.
Very close to −0.75.
Right!
3.99925 − 40.001 ≈ −0.75
Let's look back.
First: the derivative of each side.
Then: get dy/dx alone.
Last: check.
▶ Watch it solved
The question
Question: find the slope of xy = 6 at the point (2, 3).
xy = 6
y is not written alone, but y still depends on x.
So for y we use the chain rule.
y squared gives 2y, times dy/dx.
y depends on x: y2 → 2y × dydx
1 The derivative of each side
xy is x times y: use the product rule.
The derivative of x is 1, and of y is dy/dx.
(uv)′ = u′v + uv′
xy → y + x dydx
Now each term.
x terms as usual, y terms times dy/dx, a number alone gives 0.
y + x dydx = 0
2 Get dy/dx alone
Keep the dy/dx terms on the left.
Move the others to the right: they change sign.
x dydx = −y
To remove times x, we divide both sides by it.
dydx = −yx
Tidy up: dy/dx = −y over x.
dydx = −yx
3 The slope at the point
First, is (2, 3) on the curve?
Put x = 2 and y = 3 in.
We get 6.
Yes, it is.
(2)(3) = 6 ✓
Put x = 2 and y = 3 into dy/dx.
So the slope at (2, 3) is −3/2.
dydx = −32
4 Check
Let's check with a very close point on the curve.
At x = 2.001, a calculator gives y ≈ 2.9985.
x = 2.001: y ≈ 2.9985
The slope between the two points: down 0.0015, across 0.001.
About −1.5.
Very close to −1.5.
Right!
2.9985 − 30.001 ≈ −1.5
Let's look back.
First: the derivative of each side.
Then: get dy/dx alone.
Then: the slope at the point.
Last: check.
▶ Watch it solved
The question
Question: find dy/dx for x² + xy + y² = 7.
x2 + xy + y2 = 7
y is not written alone, but y still depends on x.
So for y we use the chain rule.
y squared gives 2y, times dy/dx.
y depends on x: y2 → 2y × dydx
1 The derivative of each side
xy is x times y: use the product rule.
The derivative of x is 1, and of y is dy/dx.
(uv)′ = u′v + uv′
xy → y + x dydx
Now each term.
x terms as usual, y terms times dy/dx, a number alone gives 0.
2x + y + x dydx + 2y dydx = 0
Careful!
y depends on x.
So we must multiply by dy/dx.
Careful: not y2 → 2y ✗
2 Get dy/dx alone
Keep the dy/dx terms on the left.
Move the others to the right: they change sign.
x dydx + 2y dydx = −2x − y
Two terms have dy/dx.
Take dy/dx out of both.
dydx(x + 2y) = −2x − y
To remove times (x + 2y), we divide both sides by it.
dydx = −2x − yx + 2y
So dy/dx is minus the fraction (2x + y)/(x + 2y).
dydx = −2x + yx + 2y
3 Check
Let's check at a point on the curve: (1, 2).
Our dy/dx gives −0.8 there.
at (1, 2): dy/dx = −0.8
Now a very close point on the curve.
At x = 1.001, a calculator gives y ≈ 1.9992.
x = 1.001: y ≈ 1.9992
The slope between the two points: down 0.0008, across 0.001.
About −0.8.
Very close to −0.8.
Right!
1.9992 − 20.001 ≈ −0.8
Let's look back.
First: the derivative of each side.
Then: get dy/dx alone.
Last: check.
Answer dydx = −2x + yx + 2y
▶ Watch it solved
