2.3 Q-17

Question Statement

Given the function $y = x^4 + 2x^2 + 2$, prove that:

$$\frac{dy}{dx} = 4x \sqrt{y - 1}$$

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Background and Explanation

In this problem, we need to differentiate the function $y = x^4 + 2x^2 + 2$ with respect to $x$ and prove the given identity. To solve this, we’ll use basic differentiation rules and algebraic manipulations. The key steps involve isolating terms and simplifying the expression to match the required form.

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Solution

Step 1: Express $y - 1$ in a simplified form

We start with the given function:

$$y = x^4 + 2x^2 + 2$$

Now, subtract 1 from both sides to simplify:

$$y - 1 = x^4 + 2x^2 + 1$$

This gives us:

$$y - 1 = (x^2 + 1)^2$$

Step 2: Take the square root of both sides

Now, take the square root of both sides to isolate the square root expression:

$$\sqrt{y - 1} = x^2 + 1 \tag{1}$$

This is an important equation that we’ll use in the next steps.

Step 3: Differentiate $y$ with respect to $x$

Now, differentiate the original equation $y = x^4 + 2x^2 + 2$ with respect to $x$:

$$\frac{dy}{dx} = 4x^3 + 4x$$

This gives us:

$$\frac{dy}{dx} = 4x(x^2 + 1) \tag{2}$$

Step 4: Substitute from equation (1) into equation (2)

Now, from equation (1), we know that:

$$x^2 + 1 = \sqrt{y - 1}$$

Substitute this into the derivative expression:

$$\frac{dy}{dx} = 4x \sqrt{y - 1}$$

Thus, we have proved the required identity:

$$\frac{dy}{dx} = 4x \sqrt{y - 1}$$

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Key Formulas or Methods Used

• Power Rule of Differentiation:

$$\frac{d}{dx} x^n = n x^{n-1}$$

• Square Root Property:

$$\sqrt{a} = (a)^{1/2}$$

• Substitution to simplify the derivative expression.

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Summary of Steps

1. Express $y - 1 = (x^2 + 1)^2$.
2. Take the square root of both sides: $\sqrt{y - 1} = x^2 + 1$.
3. Differentiate the original function $y$ with respect to $x$, obtaining $\frac{dy}{dx} = 4x(x^2 + 1)$.
4. Substitute $x^2 + 1 = \sqrt{y - 1}$ into the derivative expression.
5. Simplify to prove that $\frac{dy}{dx} = 4x \sqrt{y - 1}$.