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Find by implicit differentiation
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Q. Find by implicit differentiation
- Differentiate with respect to : Differentiate both sides of the equation with respect to . The left side of the equation involves the tangent of a sum, which requires the use of the chain rule for differentiation. The right side is a simple derivative of a linear term. Differentiate with respect to : Differentiate with respect to :
- Apply chain rule: Apply the chain rule to the left side of the equation.The derivative of with respect to is because is a function of and its derivative with respect to is .So, becomes
- Set derivatives equal: Set the derivatives from both sides equal to each other.
- Solve for (\frac{dy}{dx}): Solve for \((\frac{dy}{dx}).\(\newlineTo isolate (\frac{dy}{dx}), we need to move all terms not involving \((\frac{dy}{dx}) to the other side of the equation.\(\newline\sec^\(2(x+y) \cdot (\frac{dy}{dx}) = - \sec^(x+y) \cdot (\frac{dy}{dx}) = \frac{\(4\) - \(4\)\cdot\sec^\(2\)(\(4\)x+y)}{\sec^\(2\)(\(4\)x+y)}
- Simplify expression: Simplify the expression for \((\frac{dy}{dx}). We can factor out the on the right side of the equation to get:
- Apply trigonometric identity: Recognize that is equal to . This is a trigonometric identity. Applying this identity, we get:
- Correct previous mistake: Realize that there is a mistake in the previous step. The trigonometric identity used in the previous step is incorrect. The correct identity is . Therefore, the simplification made in Step is not valid. We need to correct this mistake.
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