Find the indicated derivative. d / d t ( 5 − 7 t − 3 ) d/dt(5-7t^{-3}) d / d t ( 5 − 7 t − 3 )

In this problem, we're asked to find the derivative d d t of 5 minus 7 t to the power of negative 3. So let's go ahead and jump into this derivative here. Now here, d d t tells us we're differentiating with respect to t, which makes sense because in our function here, we have the variable t. So let's go ahead and break this down. Now the derivative, d dt of 5 minus 7 t to the power of negative 3, I can break this down in each of my terms because this means that I'm taking the derivative d dt of 5, and I am then subtracting the derivative of this second term here. Since I have a constant, I can go ahead and pull that constant out to the front. So this is effectively a negative 7 times d dt of t to the power of negative 3. Now looking at this, my very first term, d dt of 5, this is a constant, meaning that my derivative here is just 0. So I have 0 minus 7 times the derivative of t to the power of negative 3. Now even though I have a negative exponent here, I can still, of course, use the power rule no matter what my power is. So using my power rule, I can multiply by that negative 3 and then decrease that exponent by 1, giving me negative 3 times t to the power of negative 3 minus 1, which is negative 4. Now I can further simplify this because this 0 is not doing anything here, and I can go ahead and multiply my negative 7 7 times negative 3 to give me a positive 21. So this is positive 21 times t to the power of negative 4. Now you may also see this rewritten as 21 over t to the power of 4 because remember that whenever you have a negative exponent, that just means that it's on the bottom of a fraction. So either of these answers are correct here for the derivative of our function, 5 minus 7 t to the power of negative 3. Let me know if you have any questions, and I'll see you in the next one.

Find the indicated derivative. d / d t ( 5 − 7 t − 3 ) d/dt(5-7t^{-3}) d / d t ( 5 − 7 t − 3 )

1 − 21 t − 4 1-21t^{-4} 1 − 21 t − 4

− 21 t − 2 -21t^{-2} − 21 t − 2

− 3 t − 4 -3t^{-4} − 3 t − 4

3. Techniques of Differentiation

Basic Rules of Differentiation

Business Calculus

3. Techniques of Differentiation

Basic Rules of Differentiation

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Multiple Choice

Find the indicated derivative.
$d/dt(5-7t^{-3})$

$1-21t^{-4}$

$-21t^{-2}$

$-3t^{-4}$

$21t^{-4}$

Verified step by step guidance

Step 1: Identify the function to differentiate. The given function is a combination of terms involving powers of t. Each term will be differentiated separately using the power rule.

Step 2: Apply the power rule for differentiation. The power rule states that if f(t) = t^n, then f'(t) = n * t^(n-1). Use this rule for each term in the function.

Step 3: Differentiate the constant term '5'. The derivative of a constant is 0, so this term contributes nothing to the derivative.

Step 4: Differentiate the term '-7t^(-3)'. Using the power rule, the derivative is (-3) * (-7) * t^(-3-1) = 21t^(-4).

Step 5: Combine the results. Since the other terms in the problem are constants or do not contribute to the derivative, the final derivative is 21t^(-4).