Using product rule,differentiate the following with respect to x:(i) y

1.	Using product rule,differentiate the following with respect to x:(i) y = x3sinx(ii) y = (x - 2)(x + 3)(iii) y = sinxcosx
Answer» (i) dy/dx = x³ d/dx(sinx) + sinx . d/dx x³ = x³cosx + sinx . 3x² = x³cosx + 3x²sinx (ii) dy/dx = (x - 2)d/dx(x + 3) (x + 3) d/dx(x - 2) = (x - 2)1 + (x + 3)1 = 2x + 1 (iii) dy/dx = sinx d/dx cosx + cosx d/dx sinx = snx . (-sinx) + cosxcosx = cos²x - sin²x

Using product rule,differentiate the following with respect to x:(i) y = x3sinx(ii) y = (x - 2)(x + 3)(iii) y = sinxcosx

Answer»

(i) dy/dx = x³ d/dx(sinx) + sinx . d/dx x³

= x³cosx + sinx . 3x²

= x³cosx + 3x²sinx

(ii) dy/dx = (x - 2)d/dx(x + 3) (x + 3) d/dx(x - 2)

= (x - 2)1 + (x + 3)1

= 2x + 1

(iii) dy/dx = sinx d/dx cosx + cosx d/dx sinx

= snx . (-sinx) + cosxcosx

= cos²x - sin²x

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