Impicit differentiation
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This document discusses techniques for finding derivatives of implicit functions. It provides examples of finding the derivative dy/dx for equations: 1) 2x + 3y = sinx 2) tan−1(x2 + y2) = a2 3) y = x sin(a + y) 4) y = √(x + √(x + √(x + − −-------)) The key steps are to differentiate both sides of the equation with respect to x and then isolate dy/dx.
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Impicit differentiation
- 2. Question 1 Find 𝑑𝑦 𝑑𝑥 if 2x + 3y = sinx Differentiate both sides w.r.t x 2 + 3 𝑑𝑦 𝑑𝑥 = cos 𝑥 𝑑𝑦 𝑑𝑥 = 𝑐𝑜𝑠𝑥 − 2 3 y is not expressed explicitly in terms of x
- 3. Question 2 Find 𝑑𝑦 𝑑𝑥 𝑤ℎ𝑒𝑛 𝑡𝑎𝑛−1 𝑥2 + 𝑦2 = 𝑎2 𝑡𝑎𝑛−1 𝑥2 + 𝑦2 = 𝑎2 𝑥2 + 𝑦2 = tan 𝑎2 Differentiate both sides w.r.t. X 2𝑥 + 2𝑦 𝑑𝑦 𝑑𝑥 = 0 y 𝑑𝑦 𝑑𝑥 = −𝑥 𝑑𝑦 𝑑𝑥 = − 𝑥 𝑦
- 4. Question 3 If y = x sin (a + y) prove that 𝑑𝑦 𝑑𝑥 = 𝑠𝑖𝑛2 (𝑎 + 𝑦) sin 𝑎 + 𝑦 − 𝑦𝑐𝑜𝑠(𝑎 + 𝑦) 𝑑𝑦 𝑑𝑥 = sin 𝑎 + 𝑦 + 𝑥 cos(𝑎 + 𝑦) 𝑑𝑦 𝑑𝑥 𝑑𝑦 𝑑𝑥 1 − 𝑥𝑐𝑜𝑠 𝑎 + 𝑦 = sin(𝑎 + 𝑦)
- 5. 𝑑𝑦 𝑑𝑥 = sin(𝑎 + 𝑦) 1 − 𝑥𝑐𝑜𝑠(𝑎 + 𝑦) = sin(𝑎 + 𝑦) 1 − 𝑦 sin 𝑎 + 𝑦 cos(𝑎 + 𝑦) 𝑑𝑦 𝑑𝑥 = 𝑠𝑖𝑛2 (𝑎 + 𝑦) sin 𝑎 + 𝑦 − 𝑦𝑐𝑜𝑠(𝑎 + 𝑦)
- 6. Question 4 If 𝑦 = √(𝑥 + √(𝑥 + 𝑥 +−−−− − 𝑦2 = 𝑥 + 𝑥 + (𝑥 +−−− − 𝑦2 = 𝑥 + 𝑦 Differentiate both sides w.r.t x 𝑦 = √(𝑥 + √(𝑥 + √(𝑥 + − −-------)
- 7. 2𝑦 𝑑𝑦 𝑑𝑥 = 1 + 𝑑𝑦 𝑑𝑥 𝑑𝑦 𝑑𝑥 2𝑦 − 1 = 1 *****************************