Step by step
The question
Question: find dy/dx for y = √(x² + 9).
1 The inside
There is something inside: x² + 9.
Call it u.
Then y is just u^(1/2).
Written as a power of u.
A square root is the power one half.
2 The outside
First the outside.
Use the power rule on u^(1/2).
We get 1/2 × u^(−1/2).
Take 1 away from the power: 1/2 − 1 = −1/2.
3 The inside's slope
Now the inside.
The power rule on x² + 9 gives 2x.
4 Multiply
The chain rule: multiply the two.
dy/du times du/dx.
Put the inside back in place of u.
Careful!
Do not stop here!
We must multiply by the inside's slope, 2x.
Multiply the numbers in front.
Then write the minus power at the bottom.
So dy/dx = x over √(x² + 9).
5 Check
Let's check: the derivative is the slope.
At x = 0, the rule gives 0.
Now the slope between two very close points: x = 0 and x = 0.001.
A calculator gives about 0.
Very close to 0.
Right!
Let's look back.
First: the inside.
Then: the outside.
Then: the inside's slope.
Then: multiply.
Last: check.