Find the third derivative of the given function. f ( x ) = 24 + 4 x 5 f(x)=24+4x^5 f ( x ) = 24 + 4 x 5

In this problem, we're asked to find the 3rd derivative of our given function, which here is f of x equals 24+4xtothepowerof5. Now remember to find the 3rd derivative, we're just doing repeated differentiation. So here we need to take the derivative three times to get that 3rd derivative. So let's start by finding our first derivative, f prime of x, which is going to give us let's see. This is a constant here, so I know that that derivative is just 0. I don't have to worry about it. And for this second term, I need to go ahead and use the power rule, multiplying by that 5, that exponent, and then decreasing that exponent by 1. This will give me 20x to the power of 4. That is our first derivative. We are, of course, not done yet because we need to keep going until we get to our 3rd derivative. Now our second derivative here, f double prime of x, again, using the power rule, taking that 4, multiplying by it in the front, decreasing that power by 1. 20 times 4 will give me 80, so I have 80 x to the power of 4 minuteus 1, which is 3. Now one final derivative that we need to take here to get to that third derivative, f triple prime of x. Taking that final derivative here, I'm going to multiply by that power. 3 times 80 will give me 240. And then, of course, decrease that power by 1. X to the power of 3 minuteus 1 gives me x squared. And here is our final answer. Our third derivative, f triple prime of x, is equal to 240x squared. Now feel free to let me know if you have any questions here, and I'll see you in the next video.

Find the third derivative of the given function. f ( x ) = 24 + 4 x 5 f(x)=24+4x^5 f ( x ) = 24 + 4 x 5

24 − 240 x 2 24-240x^2 24 − 240 x 2

3. Techniques of Differentiation

Higher Order Derivatives

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3. Techniques of Differentiation

Higher Order Derivatives

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Multiple Choice

Find the third derivative of the given function.
$f(x)=24+4x^5$

$60x^2$

$80x^3$

$240x^2$

$24-240x^2$

Verified step by step guidance

Start by identifying the given function: $ f(x) = 24 + 4x^5 $. The goal is to find the third derivative, $ f^{(3)}(x) $.

First, compute the first derivative, $ f'(x) $, by applying the power rule: $ \frac{d}{dx}[x^n] = n \cdot x^{n-1} $. For $ f(x) = 24 + 4x^5 $, the derivative of the constant $ 24 $ is $ 0 $, and the derivative of $ 4x^5 $ is $ 20x^4 $. Thus, $ f'(x) = 20x^4 $.

Next, compute the second derivative, $ f''(x) $, by differentiating $ f'(x) = 20x^4 $ again using the power rule. The derivative of $ 20x^4 $ is $ 80x^3 $. Therefore, $ f''(x) = 80x^3 $.

Now, compute the third derivative, $ f^{(3)}(x) $, by differentiating $ f''(x) = 80x^3 $ once more using the power rule. The derivative of $ 80x^3 $ is $ 240x^2 $. Thus, $ f^{(3)}(x) = 240x^2 $.

Finally, verify the result by reviewing each differentiation step to ensure no errors were made. The third derivative of $ f(x) = 24 + 4x^5 $ is $ f^{(3)}(x) = 240x^2 $.