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# 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): (ax + b)ⁿ(cx + d)ᵐ

**Solution:**

Finding derivative of (ax + b)ⁿ:

Let f(x) = (ax + b)ⁿ. It can be written as f(x) = (ax + b) (ax + b) (ax + b) ... (ax + b) *(n times)*.

Using the product rule, its derivative is,

d/dx (f(x)) = [ (ax + b) (ax + b) ... (ax + b) *(n-1) times *d/dx (ax + b)

+ (ax + b) (ax + b) ... (ax + b) *(n-1) times *d/dx (ax + b)

+ ... + (ax + b) (ax + b) ... (ax + b) *(n-1) times *d/dx (ax + b) ] *(n times)*

= (ax + b) (ax + b) ... (ax + b) *(n-1) times *[ a + a + ... + a *(n times)*]

= (ax + b)^{n-1} [na]

= na (ax + b)^{n-1}

Finding derivative of (cx + d)ᵐ:

Using the same process as above, d/dx (cx + d)ᵐ = mc (cx + d)^{m-1}

Finding derivative of given function:

Let g(x) = (ax + b)ⁿ(cx + d)ᵐ

Using the product rule, its derivative is,

d/dx (g(x)) = (ax + b)ⁿ d/dx (cx + d)ᵐ + (cx + d)ᵐ d/dx (ax + b)ⁿ

= (ax + b)ⁿ [mc (cx + d)^{m-1}] + (cx + d)ᵐ [na (ax + b)^{n-1}]

= (ax + b)^{n-1 }(cx + d)^{m-1} [mc (ax + b) + na (cx + d)]

NCERT Solutions Class 11 Maths Chapter 13 Exercise ME Question 13

## 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): (ax + b)ⁿ(cx + d)ᵐ

**Summary:**

The derivative of the given function (ax + b)ⁿ(cx + d)ᵐ is (ax + b)^{n-1 }(cx + d)^{m-1} [mc (ax + b) + na (cx + d)]

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