Ray Burns
07/10/2024 · Middle School
Let \( f(x)=\sqrt{4 x^{2}+2 x+2} \) \( f^{\prime}(x)=\square \) \( f^{\prime}(5)=\square \)
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Step-by-step Solution
Find the first order derivative with respect to \( x \) for \( \sqrt{4 x^{2}+2 x+2} \).
Evaluate the derivative by following steps:
- step0: Evaluate the derivative:
\(\frac{d}{dx}\left(\sqrt{4x^{2}+2x+2}\right)\)
- step1: Use differentiation rules:
\(\frac{d}{dg}\left(\sqrt{g}\right)\times \frac{d}{dx}\left(4x^{2}+2x+2\right)\)
- step2: Find the derivative:
\(\frac{1}{2\sqrt{g}}\times \frac{d}{dx}\left(4x^{2}+2x+2\right)\)
- step3: Find the derivative:
\(\frac{1}{2\sqrt{g}}\times \left(8x+2\right)\)
- step4: Substitute back:
\(\frac{1}{2\sqrt{4x^{2}+2x+2}}\times \left(8x+2\right)\)
- step5: Rewrite the expression:
\(\frac{1}{2\sqrt{4x^{2}+2x+2}}\times 2\left(4x+1\right)\)
- step6: Reduce the fraction:
\(\frac{1}{\sqrt{4x^{2}+2x+2}}\times \left(4x+1\right)\)
- step7: Multiply the terms:
\(\frac{4x+1}{\sqrt{4x^{2}+2x+2}}\)
- step8: Simplify:
\(\frac{4x\sqrt{4x^{2}+2x+2}+\sqrt{4x^{2}+2x+2}}{4x^{2}+2x+2}\)
Substitute \( x=5 \) into the expression \( \frac{4x\sqrt{4x^{2}+2x+2}+\sqrt{4x^{2}+2x+2}}{4x^{2}+2x+2} \).
Evaluate the expression by following steps:
- step0: Evaluate:
\(\frac{4x\sqrt{4x^{2}+2x+2}+\sqrt{4x^{2}+2x+2}}{4x^{2}+2x+2}\)
- step1: Substitute:
\(\frac{4\times 5\sqrt{4\times 5^{2}+2\times 5+2}+\sqrt{4\times 5^{2}+2\times 5+2}}{4\times 5^{2}+2\times 5+2}\)
- step2: Multiply the terms:
\(\frac{4\times 5\sqrt{100+2\times 5+2}+\sqrt{4\times 5^{2}+2\times 5+2}}{4\times 5^{2}+2\times 5+2}\)
- step3: Multiply the numbers:
\(\frac{4\times 5\sqrt{100+10+2}+\sqrt{4\times 5^{2}+2\times 5+2}}{4\times 5^{2}+2\times 5+2}\)
- step4: Add the numbers:
\(\frac{4\times 5\sqrt{112}+\sqrt{4\times 5^{2}+2\times 5+2}}{4\times 5^{2}+2\times 5+2}\)
- step5: Multiply the terms:
\(\frac{4\times 5\sqrt{112}+\sqrt{100+2\times 5+2}}{4\times 5^{2}+2\times 5+2}\)
- step6: Multiply the numbers:
\(\frac{4\times 5\sqrt{112}+\sqrt{100+10+2}}{4\times 5^{2}+2\times 5+2}\)
- step7: Simplify the root:
\(\frac{4\times 5\times 4\sqrt{7}+\sqrt{100+10+2}}{4\times 5^{2}+2\times 5+2}\)
- step8: Add the numbers:
\(\frac{4\times 5\times 4\sqrt{7}+\sqrt{112}}{4\times 5^{2}+2\times 5+2}\)
- step9: Simplify the root:
\(\frac{4\times 5\times 4\sqrt{7}+4\sqrt{7}}{4\times 5^{2}+2\times 5+2}\)
- step10: Multiply the terms:
\(\frac{80\sqrt{7}+4\sqrt{7}}{4\times 5^{2}+2\times 5+2}\)
- step11: Multiply the terms:
\(\frac{80\sqrt{7}+4\sqrt{7}}{100+2\times 5+2}\)
- step12: Multiply the numbers:
\(\frac{80\sqrt{7}+4\sqrt{7}}{100+10+2}\)
- step13: Add the numbers:
\(\frac{84\sqrt{7}}{100+10+2}\)
- step14: Add the numbers:
\(\frac{84\sqrt{7}}{112}\)
- step15: Reduce the fraction:
\(\frac{3\sqrt{7}}{4}\)
The derivative of the function \( f(x)=\sqrt{4 x^{2}+2 x+2} \) is \( f^{\prime}(x)=\frac{4x\sqrt{4x^{2}+2x+2}+\sqrt{4x^{2}+2x+2}}{4x^{2}+2x+2} \).
Substitute \( x=5 \) into the derivative expression, we get \( f^{\prime}(5)=\frac{3\sqrt{7}}{4} \).
Quick Answer
\( f^{\prime}(x)=\frac{4x\sqrt{4x^{2}+2x+2}+\sqrt{4x^{2}+2x+2}}{4x^{2}+2x+2} \)
\( f^{\prime}(5)=\frac{3\sqrt{7}}{4} \)
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