Question 62 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): x4(5sin⁡x−3cos⁡x)x^4(5 \sin x - 3 \cos x)

Correct Answer

Let f(x) = x⁴(5 sin x – 3 cos x)

Using the product rule: f'(x) = (x⁴)'(5 sin x – 3 cos x) + x⁴(5 sin x – 3 cos x)'

= 4x³(5 sin x – 3 cos x) + x⁴(5 cos x + 3 sin x)

= 20x³ sin x – 12x³ cos x + 5x⁴ cos x + 3x⁴ sin x

= x³[(20 + 3x) sin x + (5x – 12) cos x]

Exercise: Miscellaneous Exercise on Chapter 12 | Q: 22 | (Page Page 37)
For More Understanding

Explanation

This question tests the application of the product rule for differentiation. The function is a product of x⁴ and (5 sin x – 3 cos x). The derivative of x⁴ is 4x³, and the derivative of (5 sin x – 3 cos x) is (5 cos x + 3 sin x) using the standard derivatives. The product rule formula is applied: (uv)' = u'v + uv'.

Solution Steps

  1. Step 1: Identify u = x⁴ and v = (5 sin x – 3 cos x)

  2. Step 2: Find u' = 4x³ and v' = 5 cos x + 3 sin x

  3. Step 3: Apply product rule: f'(x) = u'v + uv'

  4. Step 4: Simplify: 4x³(5 sin x – 3 cos x) + x⁴(5 cos x + 3 sin x)

  5. Step 5: Expand and collect terms to get final answer

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