Solve a = b^3 ( 4 - c )

Simplify

a = b^{3} (4 - c)

Find the first partial derivative by treating the variable c as a constant and differentiating with respect to b

\frac{\partial a}{\partial b} = \frac{\partial}{\partial b} (b^{3} (4 - c))

Use differentiation rule \frac{\partial}{\partial x} (f (x) \times g (x)) = \frac{\partial}{\partial x} (f (x)) \times g (x) + f (x) \times \frac{\partial}{\partial x} (g (x))

\frac{\partial a}{\partial b} = \frac{\partial}{\partial b} (b^{3}) (4 - c) + b^{3} \times \frac{\partial}{\partial b} (4 - c)

Use \frac{\partial}{\partial x} x^{n} = n x^{n - 1} to find derivative

\frac{\partial a}{\partial b} = 3 b^{2} (4 - c) + b^{3} \times \frac{\partial}{\partial b} (4 - c)

Evaluate

\frac{\partial a}{\partial b} = 12 b^{2} - 3 c b^{2} + b^{3} \times \frac{\partial}{\partial b} (4 - c)

Use \frac{\partial}{\partial x} (c) = 0 to find derivative

\frac{\partial a}{\partial b} = 12 b^{2} - 3 c b^{2} + b^{3} \times 0

Evaluate

\frac{\partial a}{\partial b} = 12 b^{2} - 3 c b^{2} + 0

Solution

\frac{\partial a}{\partial b} = 12 b^{2} - 3 c b^{2}

Function