H2 Maths 5.1 - Product, Quotient, and Chain Rules (Combined)

Step 1 - Set up the problem

Differentiate f(x) = x²e^(3x) / ln x.

Step 1 - Set up the problem

The function is a quotient - use the quotient rule.

Step 1 - Set up the problem

The numerator x²e^(3x) is a product, so the product rule and chain rule are also needed.

Step 2 - Differentiate the numerator

Let u = x²e^(3x).

Step 2 - Differentiate the numerator

By the product rule: u' = (x²)' · e^(3x) + x² · (e^(3x))'.

Step 2 - Differentiate the numerator

Chain rule: d/dx[e^(3x)] = 3e^(3x).

Step 2 - Differentiate the numerator

So u' = 2xe^(3x) + 3x²e^(3x) = e^(3x)(2x + 3x²).

Step 3 - Differentiate the denominator

Let v = ln x.

Step 3 - Differentiate the denominator

v' = 1/x.

Step 4 - Apply the quotient rule

Quotient rule: f'(x) = (u'v − uv') / v².

Step 4 - Apply the quotient rule

Substituting into the formula.

Step 4 - Apply the quotient rule

The second term: x² · (1/x) = x. So the numerator becomes e^(3x)(2x + 3x²) ln x minus xe^(3x).

Step 5 - Factor and write the final answer

Both numerator terms share a factor of xe^(3x).

Step 5 - Factor and write the final answer

Factoring gives xe^(3x)[(2 + 3x)ln x − 1].

Step 5 - Factor and write the final answer

That is the final answer.