Find the derivative of the following via implicit differentiation:
d/dx(f(x)) = d/dx(4 (11+x^3))
The derivative of f(x) is f'(x):
f'(x) = d/dx(4 (11+x^3))
Factor out constants:
f'(x) = 4 (d/dx(11+x^3))
Differentiate the sum term by term:
f'(x) = 4 d/dx(11)+d/dx(x^3)
The derivative of 11 is zero:
f'(x) = 4 (d/dx(x^3)+0)
Simplify the expression:
f'(x) = 4 (d/dx(x^3))
Use the power rule, d/dx(x^n) = n x^(n-1), where n = 3: d/dx(x^3) = 3 x^2:
f'(x) = 4 3 x^2
Simplify the expression:
f'(x) = 12 x^2
Expand the left hand side:
Answer: |f'(x) = 12 x^2
