Step by step
The question
Question: find dy/dx for y = x³ˣ, using logarithms.
x is in the base and in the power.
The power rule needs a fixed power.
The bˣ rule needs a fixed base.
Neither works, so we take ln.
Careful!
The power rule is only for a fixed power.
Here the power changes with x.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Take ln of both sides
Take ln of both sides.
Both sides stay equal.
2 Use the log rules
The log rule: a power inside ln comes down in front.
So ln y is 3x times ln x.
No more power!
3 The derivative of each side
The left side: y depends on x, so the chain rule gives 1/y, times dy/dx.
The right side is a product: use the product rule.
3x gives 3.
And ln x gives 1 over x.
4 Multiply by y
Multiply both sides by y, to get dy/dx alone.
5 Put y back
Tidy the second part: 3x times 1/x is 3.
Put y = x³ˣ back: dy/dx = x³ˣ(3 ln x + 3).
6 Check
Let's check at one x.
Any x works above 0: take x = 2.
The derivative is the slope there.
Our answer gives 325.0843, with a calculator.
Now the slope between two very close points: x = 2 and x = 2.001.
A calculator gives about 325.96.
Very close to 325.0843.
Right!
7 Pick the choice
Our answer is dy over dx = x³ˣ(3 ln x + 3).
That is choice B.
The same brackets, in the other order: in times, the order does not matter.
8 Why not the others
A is 3x × x^(3x − 1).
Wrong: the power rule needs a number power.
C is x³ˣln x.
Wrong: lost the 3 and the product rule.
D is 3x³ˣln x.
Wrong: lost the + 3 from the product rule.
Let's look back.
First: take ln of both sides.
Then: use the log rules.
Then: the derivative of each side.
Then: multiply by y.
Then: put y back.
Then: check.
Then: pick the choice.
Last: why not the others.