Question Statement

Find the differential $d y$ for the function $y = x^{2} + 1$ .

Background and Explanation

Differentials provide a linear approximation of how a function changes with small changes in its input. For a function $y = f (x)$ , the differential is defined as $d y = f^{'} (x) d x$ , representing the principal (linear) part of the increment $Δ y$ . This concept extends to approximating function values near known points using the formula $f (x + Δ x) \approx f (x) + f^{'} (x) Δ x$ .

Solution

Part (a): Differential of $y = x^{2} + 1$

Given the function: $y = x^{2} + 1$

First, we examine the increment $Δ y$ when $x$ changes by $Δ x$ : $y + Δ y = (x + Δ x)^{2} + 1$

Expanding and solving for $Δ y$ : $Δ y = (x + Δ x)^{2} + 1 - y = x^{2} + Δ x^{2} + 2 x Δ x + 1 - (x^{2} + 1) = x^{2} + Δ x^{2} + 2 x Δ x + 1 - x^{2} - 1 = Δ x^{2} + 2 x Δ x = Δ x (Δ x + 2 x)$

For the differential, we differentiate $y = x^{2} + 1$ with respect to $x$ : $d y = d (x^{2} + 1) = d (x^{2}) + d (1) = 2 x d x + 0 = 2 x d x$

Key Formulas or Methods Used

Definition of Differential: $d y = f^{'} (x) d x$
Increment Method: $Δ y = f (x + Δ x) - f (x)$
Trigonometric Identity: $sin A - sin B = 2 cos (\frac{A + B}{2}) sin (\frac{A - B}{2})$
Linear Approximation Formula: $f (x + Δ x) \approx f (x) + f^{'} (x) Δ x$
Power Rule for Differentiation: $\frac{d}{d x} (x^{n}) = n x^{n - 1}$

Summary of Steps

For $y = x^{2} + 1$ : Calculate $Δ y = (x + Δ x)^{2} + 1 - (x^{2} + 1) = Δ x (Δ x + 2 x)$ , then find $d y = 2 x d x$

Question Statement

Find the differential $d y$ for the function $y = x^{2} + 1$ .

Background and Explanation

Solution

Part (a): Differential of $y = x^{2} + 1$

Given the function: $y = x^{2} + 1$

First, we examine the increment $Δ y$ when $x$ changes by $Δ x$ : $y + Δ y = (x + Δ x)^{2} + 1$

For the differential, we differentiate $y = x^{2} + 1$ with respect to $x$ : $d y = d (x^{2} + 1) = d (x^{2}) + d (1) = 2 x d x + 0 = 2 x d x$

Key Formulas or Methods Used

Definition of Differential: $d y = f^{'} (x) d x$
Increment Method: $Δ y = f (x + Δ x) - f (x)$
Trigonometric Identity: $sin A - sin B = 2 cos (\frac{A + B}{2}) sin (\frac{A - B}{2})$
Linear Approximation Formula: $f (x + Δ x) \approx f (x) + f^{'} (x) Δ x$
Power Rule for Differentiation: $\frac{d}{d x} (x^{n}) = n x^{n - 1}$

Summary of Steps

For $y = x^{2} + 1$ : Calculate $Δ y = (x + Δ x)^{2} + 1 - (x^{2} + 1) = Δ x (Δ x + 2 x)$ , then find $d y = 2 x d x$

2.7 Q-27

Question Statement

Background and Explanation

Solution

Part (a): Differential of $y = x^{2} + 1$

Key Formulas or Methods Used

Summary of Steps

2.7 Q-27

Question Statement

Background and Explanation

Solution

Part (a): Differential of $y = x^{2} + 1$

Key Formulas or Methods Used

Summary of Steps

Question Statement

Background and Explanation

Solution

Part (a): Differential of y=x2+1

Key Formulas or Methods Used

Summary of Steps

Question Statement

Background and Explanation

Solution

Part (a): Differential of y=x2+1

Key Formulas or Methods Used

Summary of Steps

Part (a): Differential of $y = x^{2} + 1$

Part (a): Differential of $y = x^{2} + 1$