# Video: APCALC04AB-P1A-Q10-542124916098

If 𝑓(𝑥) = (3𝑥⁴ + 16)(∛(𝑥 + 4𝑥)), find 𝑓′(𝑥).

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### Video Transcript

If 𝑓 of 𝑥 equals three 𝑥 to the fourth power plus 16 times the cube root of 𝑥 plus four 𝑥, find 𝑓 prime of 𝑥.

𝑓 of 𝑥 is itself the product of two differentiable functions, the first one being three 𝑥 to the fourth power plus 16 and the second being the cube root of 𝑥 plus four 𝑥. This means we can use the product rule to find its derivative, 𝑓 prime of 𝑥.

This says that the derivative of the product of two differentiable functions 𝑢 and 𝑣 is 𝑢 times d𝑣 by d𝑥 plus 𝑣 times d𝑢 by d𝑥. We’ll define 𝑢 as our first function, three 𝑥 to the fourth power plus 16. Then, we’ll define 𝑣 as our second function, changing the cube root of 𝑥 to 𝑥 the power of one-third. So, 𝑣 is equal to 𝑥 to the power of one-third plus four 𝑥.

We can see from the formula for the product rule that we’re going to need to differentiate each of these with respect to 𝑥. And we recall that to differentiate a polynomial term, as all of these are, we multiply the entire term by the exponent and then simply reduce that exponent by one. The derivative, then, of three 𝑥 to the fourth power is four times three 𝑥 cubed. And the derivative of 16, a constant, is just zero.

So, d𝑢 by d𝑥 simplifies to 12𝑥 cubed. The derivative of 𝑥 to the power of one-third is a third times 𝑥 to the power of negative two-thirds. And then, the derivative of four 𝑥 is simply four. Let’s replace 𝑢, d𝑣 by d𝑥, 𝑣, and d𝑢 by d𝑥 in our formula for the derivative of 𝑢 times 𝑣. It’s 𝑢 times d𝑣 by d𝑥 plus 𝑣 times d𝑢 by d𝑥.

We then convert each expression with a fractional index back into root four. And we see that 𝑥 to the power of negative two-thirds is one over the cube root of 𝑥 squared, whereas 𝑥 to the power of one-third is simply the cube root of 𝑥. And in doing so, we found 𝑓 prime of 𝑥. It’s one over three times the cube root of 𝑥 squared plus four times three 𝑥 to the fourth power plus 16 plus 12𝑥 cubed times four 𝑥 plus the cube root of 𝑥.