Ortega Bird
04/12/2024 · Junior High School
Compute the first three derivatives of \( f(t)=\sqrt{12 t^{2}+11} \) \( f^{\prime}(t)= \) \( f^{\prime \prime}(t)= \) \( f^{\prime \prime \prime}(t)= \)
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Step-by-step Solution
To compute the first three derivatives of the function \( f(t) = \sqrt{12t^2 + 11} \), we will use the chain rule and the power rule.
### First Derivative \( f'(t) \)
1. Rewrite the function:
\[
f(t) = (12t^2 + 11)^{1/2}
\]
2. Apply the chain rule:
\[
f'(t) = \frac{1}{2}(12t^2 + 11)^{-1/2} \cdot (24t)
\]
3. Simplify:
\[
f'(t) = \frac{24t}{2\sqrt{12t^2 + 11}} = \frac{12t}{\sqrt{12t^2 + 11}}
\]
### Second Derivative \( f''(t) \)
To find the second derivative, we differentiate \( f'(t) \):
\[
f'(t) = \frac{12t}{\sqrt{12t^2 + 11}}
\]
Using the quotient rule:
\[
f''(t) = \frac{(12)(\sqrt{12t^2 + 11}) - (12t) \left(\frac{1}{2}(12t^2 + 11)^{-1/2} \cdot 24t\right)}{(12t^2 + 11)}
\]
Simplifying the second term:
\[
= \frac{12\sqrt{12t^2 + 11} - 12t \cdot \frac{12t}{\sqrt{12t^2 + 11}}}{12t^2 + 11}
\]
\[
= \frac{12\sqrt{12t^2 + 11} - \frac{144t^2}{\sqrt{12t^2 + 11}}}{12t^2 + 11}
\]
\[
= \frac{12(12t^2 + 11) - 144t^2}{(12t^2 + 11)\sqrt{12t^2 + 11}}
\]
\[
= \frac{12(11 - 132t^2)}{(12t^2 + 11)\sqrt{12t^2 + 11}}
\]
Thus, the second derivative is:
\[
f''(t) = \frac{12(11 - 12t^2)}{(12t^2 + 11)^{3/2}}
\]
### Third Derivative \( f'''(t) \)
Now we differentiate \( f''(t) \):
\[
f''(t) = \frac{12(11 - 12t^2)}{(12t^2 + 11)^{3/2}}
\]
Using the quotient rule again:
\[
f'''(t) = \frac{(0 - 12 \cdot 24t)(12t^2 + 11)^{3/2} - 12(11 - 12t^2) \cdot \frac{3}{2}(12t^2 + 11)^{1/2} \cdot 24t}{(12t^2 + 11)^3}
\]
This simplifies to:
\[
= \frac{-288t(12t^2 + 11)^{3/2} - 12(11 - 12t^2)(36t(12t^2 + 11)^{1/2})}{(12t^2 + 11)^3}
\]
Combining the terms:
\[
= \frac{-288t(12t^2 + 11) - 432t(11 - 12t^2)}{(12t^2 + 11)^{5/2}}
\]
Thus, the third derivative is:
\[
f'''(t) = \frac{-288t(12t^2 + 11) - 432t(11 - 12t^2)}{(12t^2 + 11)^{5/2}}
\]
### Summary of Derivatives
\[
f'(t) = \frac{12t}{\sqrt{12t^2 + 11}}
\]
\[
f''(t) = \frac{12(11 - 12t^2)}{(12t^2 + 11)^{3/2}}
\]
\[
f'''(t) = \frac{-288t(12t^2 + 11) - 432t(11 - 12t^2)}{(12t^2 + 11)^{5/
Quick Answer
\( f^{\prime}(t)= \frac{12t}{\sqrt{12t^2 + 11}} \)
\( f^{\prime \prime}(t)= \frac{12(11 - 12t^2)}{(12t^2 + 11)^{3/2}} \)
\( f^{\prime \prime \prime}(t)= \frac{-288t(12t^2 + 11) - 432t(11 - 12t^2)}{(12t^2 + 11)^{5/2}} \)
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