Math / Calculus

QuestionQ5. Find $\frac{d y}{d x}$ when $2 x^{2}-y^{2}=5 x \sin (y)+2 a^{13}$

Studdy Solution

STEP 1

1. The equation $2x^2 - y^2 = 5x \sin(y) + 2a^{13}$ involves both $x$ and $y$ , indicating implicit differentiation is needed.
2. We assume $y$ is a function of $x$ , i.e., $y = y(x)$ .
3. The derivative of $a^{13}$ with respect to $x$ is zero since $a$ is treated as a constant.

STEP 2

1. Differentiate both sides of the equation with respect to $x$ .
2. Solve for $\frac{dy}{dx}$ .

STEP 3

Differentiate both sides of the equation with respect to $x$ :
The left side: - Differentiate $2x^2$ with respect to $x$ : $ \frac{d}{dx}(2x^2) = 4x \]
- Differentiate $-y^2$ with respect to $x$ using the chain rule: $ \frac{d}{dx}(-y^2) = -2y \frac{dy}{dx} \]
The right side: - Differentiate $5x \sin(y)$ using the product rule: $ \frac{d}{dx}(5x \sin(y)) = 5 \sin(y) + 5x \cos(y) \frac{dy}{dx} \]
- Differentiate $2a^{13}$ with respect to $x$ : $ \frac{d}{dx}(2a^{13}) = 0 \]

STEP 4

Combine the derivatives from Step 1 into a single equation:
$4x - 2y \frac{dy}{dx} = 5 \sin(y) + 5x \cos(y) \frac{dy}{dx}$

STEP 5

Rearrange the equation to solve for $\frac{dy}{dx}$ :
1. Move all terms involving $\frac{dy}{dx}$ to one side of the equation: $ -2y \frac{dy}{dx} - 5x \cos(y) \frac{dy}{dx} = 5 \sin(y) - 4x \]
2. Factor out $\frac{dy}{dx}$ : $ \frac{dy}{dx}(-2y - 5x \cos(y)) = 5 \sin(y) - 4x \]
3. Solve for $\frac{dy}{dx}$ : $ \frac{dy}{dx} = \frac{5 \sin(y) - 4x}{-2y - 5x \cos(y)} \]
The derivative $\frac{dy}{dx}$ is:
$\frac{dy}{dx} = \frac{5 \sin(y) - 4x}{-2y - 5x \cos(y)}$

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