Find the third derivative of the given function. f ( t ) = 5 t − 4 f(t)=5t^{-4} f ( t ) = 5 t − 4

Here, we're asked to find the 3rd derivative of the given function, and our given function here is f of t equals 5 t to the power of negative 4. Now to find that third derivative, we need to take the derivative three times. So let's start by finding that first derivative. Now I'm gonna denote my derivative using the prime notation, but remember that you could also use different notations to denote these derivatives and they would still be correct. So I'm going to use f prime of t here, but I also could denote this first derivative as d f d t. Now using that prime notation f prime of t, my first derivative, I'm going to use the power rule here, taking that 4 negative 4 multiplying by it in the front, and then decreasing that power by 1. Since I have that negative 4 times 5, that's going to give me negative 20 times t to the power of negative 5, taking that negative 4 minus 1 or negative 5. Now to find my next derivative, my second derivative here using the power rule yet again, taking that negative 5 multiplying by it in the front, Negative 20 times negative 5 is going to give me a positive 100 since I have 2 negatives being multiplied. And then decreasing that power by 1 t to the power of negative 5 minus 1 gives me negative 6. Now one final derivative because we wanna take our 3rd derivative here. So to take that 3rd derivative, f triple prime of t, this is going to be equal to, again, using the power rule, multiplying by that negative 6 in the front, back to a negative. We have a negative 600 times t to the power of negative 6 minus 1, which is negative 7, and this is our final answer here. Now remember that whenever you have a negative exponent, you could also rewrite this to get rid of that negative exponent by writing this as negative 600 over t to the power of 7 because that negative just tells you that that term should go on the bottom with a positive exponent. So either of these answers would be correct, but here our 3rd derivative f triple prime of t is negative 600 t to the power of negative 7. Let me know if you have any questions here, and I'll see you in the next one.

3. Techniques of Differentiation

Higher Order Derivatives

Calculus

3. Techniques of Differentiation

Higher Order Derivatives

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Multiple Choice

Find the third derivative of the given function.
$f(t)=5t^{-4}$

$0$

$120t$

$600t^7$

$- 600 t^{- 7}$

Verified step by step guidance

Start by identifying the function you need to differentiate: $ f(t) = 5t^{-4} $. This function is in the form of a power function, which is suitable for applying the power rule for differentiation.

Apply the power rule for differentiation to find the first derivative. The power rule states that if $ f(t) = at^n $, then $ f'(t) = a \cdot n \cdot t^{n-1} $. For $ f(t) = 5t^{-4} $, the first derivative $ f'(t) $ will be $ 5 \cdot (-4) \cdot t^{-5} $.

Next, find the second derivative by differentiating $ f'(t) = -20t^{-5} $ using the power rule again. The second derivative $ f''(t) $ will be $ -20 \cdot (-5) \cdot t^{-6} $.

Proceed to find the third derivative by differentiating $ f''(t) = 100t^{-6} $. Apply the power rule once more: $ f'''(t) = 100 \cdot (-6) \cdot t^{-7} $.

Simplify the expression for the third derivative: $ f'''(t) = -600t^{-7} $. This is the third derivative of the function $ f(t) = 5t^{-4} $.