Video: APCALC04AB-P1A-Q10-542124916098

If 𝑓(π‘₯) = (3π‘₯⁴ + 16)(βˆ›(π‘₯ + 4π‘₯)), find 𝑓′(π‘₯).

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Video Transcript

If 𝑓 of π‘₯ equals three π‘₯ to the fourth power plus 16 times the cube root of π‘₯ plus four π‘₯, find 𝑓 prime of π‘₯.

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

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

We can see from the formula for the product rule that we’re going to need to differentiate each of these with respect to π‘₯. 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 π‘₯ to the fourth power is four times three π‘₯ cubed. And the derivative of 16, a constant, is just zero.

So, d𝑒 by dπ‘₯ simplifies to 12π‘₯ cubed. The derivative of π‘₯ to the power of one-third is a third times π‘₯ to the power of negative two-thirds. And then, the derivative of four π‘₯ is simply four. Let’s replace 𝑒, d𝑣 by dπ‘₯, 𝑣, and d𝑒 by dπ‘₯ in our formula for the derivative of 𝑒 times 𝑣. It’s 𝑒 times d𝑣 by dπ‘₯ plus 𝑣 times d𝑒 by dπ‘₯.

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

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