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Basic Differentiation Rule

Assertion A: ...

Question

Assertion (A): The $n^{t h}$ derivative of $sin 5 x cos 3 x$ is $\frac{1}{2} [8^{n} sin (\frac{n π}{2} + 8 x) + 2^{n} sin (\frac{n π}{2} + 2 x)]$
Reason (R): lf $y = sin (a x + b)$ then $y_{n} = a^{n} sin (a x + b + \frac{n π}{2})$

A

Both A and R are true and R is the correct explanation of A

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B

Both A and R are true but R is not correct explanation of A

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C

A is true but R is false

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D

A is false but R is true

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Solution

The correct option is A Both A and R are true and R is the correct explanation of A
$L e t u s c o n s i d e r cos x T h e f i r s t d e r i v a t i v e g e t s u - s i n x . T h e s e c o n d d e r i v a t i v e b r i n g s u - c o s x . S o i n g e n e r a l t h e d e r i v a t i v e f o r m g e t s t o b e c o s (\frac{n π}{2} + x); n b e l o n g s t o i n t e g e r s o f c o u r s e . S i m i l a r l y t h e n d e r a v a t i v e o f s i n x w o u l d b e : f^{(n)} (x) = sin (\frac{n π}{2} + x) T h e r e f o r e t h e g i v e n a s s e r t i o n o f t h e t h e s t a t e m e n t a n d t h e r e a s o n w o u l d b e c o r r e c t e x p l a i n a t i o n t o t h e g i v e n a s s e r t i o n$

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