Question 53 of 70intermediate🔧 ApplyShort Answer1 mark

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,ra, b, c, d, p, q, r and ss are fixed non-zero constants and mm and nn are integers): (ax+b)n(cx+d)m(ax + b)^n(cx + d)^m

Correct Answer

an(ax+b)n−1(cx+d)m+cm(ax+b)n(cx+d)m−1an(ax + b)^{n-1}(cx + d)^m + cm(ax + b)^n(cx + d)^{m-1}

Exercise: Miscellaneous Exercise on Chapter 12 | Q: 13 | (Page Page 37)
For More Understanding

Explanation

This question requires applying the product rule combined with the chain rule. The context provides the product rule formula: d/dx(u·v) = u'·v + u·v'. For functions like (ax + b)^n, we use the chain rule: the derivative is n(ax + b)^(n-1) times the derivative of the inner function (ax + b), which is a. Similarly for (cx + d)^m, the derivative is cm(cx + d)^(m-1).

Solution Steps

  1. Step 1: Let u = (ax + b)^n and v = (cx + d)^m

  2. Step 2: Find du/dx = n(ax + b)^(n-1) · a = an(ax + b)^(n-1)

  3. Step 3: Find dv/dx = m(cx + d)^(m-1) · c = cm(cx + d)^(m-1)

  4. Step 4: Apply product rule: d/dx(uv) = (du/dx)·v + u·(dv/dx)

  5. Step 5: = an(ax + b)^(n-1)(cx + d)^m + cm(ax + b)^n(cx + d)^(m-1)

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