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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