Question

How do you differentiate `f(x)=lnx^2*e^(x^3-1)` using the product rule?

Answer 1

Use the chain rule in conjunction with the product rule.

Explanation:

We can start by using the chain rule to take the derivative of our first term, `lnx^2`. We take the derivative of the "outside," then multiply it by the derivative of the "inside." Remember that the derivative of the natural log is `1/x`.

Thus, we get `(2ln(x))/x`.

By the product rule, since we've taken the derivative of the first term, we multiply by the second term, which we leave alone.

This gives us, so far: `(2ln(x))/x*e^(x^3-1)`.

Now we do the same thing to the other term. We take the derivative of `e^(x^3-1)` using the chain rule, and multiply by the original of the first term, giving us `3lnx^2e^(x^3-1)x^2`.

By the product rule, we then add these two resulting terms together.

`(2ln(x)e^(x^3-1))/x + 3lnx^2e^(x^3-1)x^2`.

This can be further simplified as:

`(e^(x^3-1)ln(x)(3x^3ln(x)+2))/x`