Allan Watkins
03/29/2023 · Senior High School
Find the value of the derivative for the given function. \[ \left.\frac{d r}{d \theta}\right|_{\theta=2} \text { if } r=\frac{4}{\sqrt{11-\theta}} \] \( \left.\frac{d r}{d \theta}\right|_{\theta=2}=\square \) (Type an integer or a simplified fraction.)
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Find the first order derivative with respect to \( \theta \) for \( \frac{4}{\sqrt{11-\theta}} \).
Evaluate the derivative by following steps:
- step0: Evaluate the derivative:
\(\frac{d}{d\theta}\left(\frac{4}{\sqrt{11-\theta }}\right)\)
- step1: Use differentiation rules:
\(\frac{\frac{d}{d\theta}\left(4\right)\sqrt{11-\theta }-4\times \frac{d}{d\theta}\left(\sqrt{11-\theta }\right)}{\left(\sqrt{11-\theta }\right)^{2}}\)
- step2: Find the derivative:
\(\frac{0\times \sqrt{11-\theta }-4\left(-\frac{1}{2\sqrt{11-\theta }}\right)}{\left(\sqrt{11-\theta }\right)^{2}}\)
- step3: Calculate:
\(\frac{0-\left(-\frac{2}{\sqrt{11-\theta }}\right)}{11-\theta }\)
- step4: Calculate:
\(\frac{\frac{2}{\sqrt{11-\theta }}}{11-\theta }\)
- step5: Multiply by the reciprocal:
\(\frac{2}{\sqrt{11-\theta }}\times \frac{1}{11-\theta }\)
- step6: Multiply the terms:
\(\frac{2}{\sqrt{11-\theta }\times \left(11-\theta \right)}\)
- step7: Multiply the terms:
\(\frac{2}{\left(11-\theta \right)\sqrt{11-\theta }}\)
- step8: Simplify:
\(\frac{2\sqrt{11-\theta }}{\left(11-\theta \right)\left(11-\theta \right)}\)
- step9: Multiply the terms:
\(\frac{2\sqrt{11-\theta }}{121-22\theta +\theta ^{2}}\)
Substitute \( \theta=2 \) into the expression \( \frac{2\sqrt{11-\theta }}{121-22\theta +\theta ^{2}} \).
Evaluate the expression by following steps:
- step0: Evaluate:
\(\frac{2\sqrt{11-\theta }}{121-22\theta +\theta ^{2}}\)
- step1: Substitute:
\(\frac{2\sqrt{11-2}}{121-22\times 2+2^{2}}\)
- step2: Subtract the numbers:
\(\frac{2\sqrt{9}}{121-22\times 2+2^{2}}\)
- step3: Simplify the root:
\(\frac{2\times 3}{121-22\times 2+2^{2}}\)
- step4: Multiply the numbers:
\(\frac{2\times 3}{121-44+2^{2}}\)
- step5: Multiply the numbers:
\(\frac{6}{121-44+2^{2}}\)
- step6: Calculate:
\(\frac{6}{81}\)
- step7: Reduce the fraction:
\(\frac{2}{27}\)
The value of the derivative \( \left.\frac{d r}{d \theta}\right|_{\theta=2} \) for the given function \( r=\frac{4}{\sqrt{11-\theta}} \) is \( \frac{2}{27} \) or approximately \( 0.\dot{0}7\dot{4} \).
Quick Answer
The value of the derivative at \( \theta=2 \) is \( \frac{2}{27} \).
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