Question

If $y=2+\sqrt{u}$ and $u=x^{3}+1$, then $\frac{dy}{dx}=$

• A. $\frac{x^{2}}{2\sqrt{x^{3}+1}}$
• B. $\frac{3x^{2}}{\sqrt{x^{3}+1}}$
• C. $\frac{3x^{2}}{2\sqrt{x^{3}+1}}$
• D. $3x^{2}\sqrt{x^{3}+1}$
• E. $x^{2}\sqrt{x^{3}+1}$

Hint:

Calculus Tip: Alternatively, you can just substitute $u$ into $y$ immediately to get $y = 2 + \sqrt{x^3 + 1}$ and take the derivative directly using the standard chain rule. Both methods yield the exact same result!

Answer 1

The Correct Option is C

Concept:
When a function $y$ is defined in terms of an intermediate variable $u$, which is in turn defined in terms of $x$, the Chain Rule states that $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$. We must differentiate each part separately and then multiply them.

Step 1: Differentiate y with respect to u.
Given $y = 2 + u^{1/2}$: $$\frac{dy}{du} = 0 + \frac{1}{2}u^{-1/2}$$ $$\frac{dy}{du} = \frac{1}{2\sqrt{u}}$$

Step 2: Differentiate u with respect to x.
Given $u = x^3 + 1$: $$\frac{du}{dx} = 3x^2 + 0$$ $$\frac{du}{dx} = 3x^2$$

Step 3: Apply the Chain Rule.
Multiply the two derivatives together to find $\frac{dy}{dx}$: $$\frac{dy}{dx} = \left(\frac{1}{2\sqrt{u}}\right) \cdot (3x^2)$$

Step 4: Substitute x back into the expression.
Replace the intermediate variable $u$ with its original definition in terms of $x$ ($u = x^3 + 1$): $$\frac{dy}{dx} = \frac{3x^2}{2\sqrt{x^3 + 1}}$$

Step 5: Verify the final form.
The resulting expression requires no further simplification and directly matches option C. Hence the correct answer is (C) $\frac{3x^{2{2\sqrt{x^{3}+1}}$}.