Answer:
There are two interpretations.
①:
log 2 [(2x + 1)^3]
②:
[log 2 (2x + 1)]^3
①:
d/dx log 2 [(2x + 1)^3]
= 1/{[(2x + 1)^3] ln (2)}
= [3(2x + 1)^2]/{[(2x + 1)^3] ln (2)}
= [2 * 3(2x + 1)^2]/{[(2x + 1)^3] ln (2)}
= [6(2x + 1)^2]/{[(2x + 1)^3] ln (2)}
②:
d/dx [log 2 (2x + 1)]^3
= 3[log 2 (2x + 1)]^2
= {3[log 2 (2x + 1)]^2}/[(2x + 1) ln (2)]
= {2 * 3[log 2 (2x + 1)]^2}/[(2x + 1) ln (2)]
= {6[log 2 (2x + 1)]^2}/[(2x + 1) ln (2)]
Explanation:
Chain rule:
d/dx f(g(h... w(x)))
= f’(g(h(... w(x)))) * g’(h(... w(x))) * h’(... w(x)) * ... * w’(x)
You derive functions inside another function by differentiating the outer function while leaving the insides intact, then multiplying with the derivative of the function inside it while leaving the insides intact, and so on and so on until the innermost one.
Derivatives of the functions:
d/dx x^n = nx^(n - 1)
d/dx log n (x) = 1/[x ln (n)]