Question 63 of 70intermediate🔧 ApplyShort Answer1 mark

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x2+1)cos⁡x(x^2 + 1) \cos x

Correct Answer

Let f(x)=(x2+1)cos⁡xf(x) = (x^2 + 1) \cos x. Using the product rule:

ddx[(x2+1)cos⁡x]=(x2+1)′⋅cos⁡x+(x2+1)⋅(cos⁡x)′\frac{d}{dx}[(x^2 + 1) \cos x] = (x^2 + 1)' \cdot \cos x + (x^2 + 1) \cdot (\cos x)' =2x⋅cos⁡x+(x2+1)⋅(−sin⁡x)= 2x \cdot \cos x + (x^2 + 1) \cdot (-\sin x) =2xcos⁡x−(x2+1)sin⁡x= 2x \cos x - (x^2 + 1) \sin x
Exercise: Miscellaneous Exercise on Chapter 12 | Q: 23 | (Page Page 37)
For More Understanding

Explanation

This question requires applying the product rule for differentiation, which is demonstrated in Example 18 of the textbook context. The product rule states that for two functions u(x) and v(x), the derivative of their product is u'(x)·v(x) + u(x)·v'(x). Here, u(x) = x² + 1 has derivative 2x, and v(x) = cos x has derivative -sin x (as shown in the quotient rule derivation in the context where (cos x)' appears as -sin x).

Solution Steps

  1. Step 1: Identify the function as a product of (x² + 1) and cos x

  2. Step 2: Apply product rule: derivative = (x² + 1)'·cos x + (x² + 1)·(cos x)'

  3. Step 3: Calculate derivatives: (x² + 1)' = 2x and (cos x)' = -sin x

  4. Step 4: Substitute: 2x·cos x + (x² + 1)·(-sin x) = 2x cos x - (x² + 1) sin x

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