Question:
If $\frac{dy}{dx} = \frac{1}{8\left(\sqrt{16+\sqrt{25+\sqrt{x}}}\right)\left(\sqrt{25+\sqrt{x}}\right)\sqrt{x}}$, then $y =$
If $\frac{dy}{dx} = \frac{1}{8\left(\sqrt{16+\sqrt{25+\sqrt{x}}}\right)\left(\sqrt{25+\sqrt{x}}\right)\sqrt{x}}$, then $y =$
Show Hint
Look for nested square roots; the derivative of $\sqrt{f(x)}$ is $\frac{f'(x)}{2\sqrt{f(x)}}$.
Updated On: Apr 28, 2026
- $\sqrt{16+\sqrt{25+\sqrt{x}}}+C$
- $\sqrt{16+\sqrt{25+\sqrt{x}}}+x+C$
- $\sqrt{16+\sqrt{25+\sqrt{x}}}+x^{2}+C$
- $x\sqrt{16+\sqrt{25+\sqrt{x}}}+C$
- $x^{2}\sqrt{16+\sqrt{25+\sqrt{x}}}+C$
Show Solution
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The Correct Option is A
Solution and Explanation
Step 1: Concept
Recognize the chain rule pattern. $y$ is the integral of the given expression.
Step 2: Analysis
Let $u = \sqrt{16+\sqrt{25+\sqrt{x}}}$. Differentiating $u$ using the chain rule matches the integrand.
Step 3: Conclusion
The integral of the derivative of $f(x)$ is $f(x) + C$. Therefore, $y = \sqrt{16+\sqrt{25+\sqrt{x}}} + C$. Final Answer: (A)
Recognize the chain rule pattern. $y$ is the integral of the given expression.
Step 2: Analysis
Let $u = \sqrt{16+\sqrt{25+\sqrt{x}}}$. Differentiating $u$ using the chain rule matches the integrand.
Step 3: Conclusion
The integral of the derivative of $f(x)$ is $f(x) + C$. Therefore, $y = \sqrt{16+\sqrt{25+\sqrt{x}}} + C$. Final Answer: (A)
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