How to find the derivative of $A=3x^2 (25-2x)^5$

Question

For $A=3x^2(25-2x)^5$, I need to show the steps on how to get the derivative. The derivative provided is $6x(25-2x)^4(25-7x)$.

The equation is formed from $A=3x^2y$ and $y=(25-2x)^5$ from a differentiation question.

Answer 1

When you face products, quotients and powers, logarithmic differentiation use to make life easier $$A=3x^2 (25-2x)^5\implies \log(A)=\log(3)+2\log(x)+5\log(25-2x)$$ $$\frac {A'}A=\frac 2x-\frac {10}{25-2x}=\frac {2(25-7x)}{x(25-2x)}$$ $$A' =A \times \frac {A'}A=3x^2(25-2x)^5 \frac {2(25-7x)}{x(25-2x)}=6 x(25-7 x) (25-2 x)^4 $$

Answer 2

HINT

1. Find the derivative of $f(x) = 3x^2$ using the chain rule
2. Find the derivative of $g(x) = (25-2x)^5$ using the chain rule
3. Find the derivative of $f(x)g(x)$ using the product rule.

Answer 3

Group the terms to get the final answer:

$$6x(25-2x)^5-10(25-2x)^4 \cdot 3x^2$$ $$=6x \left((25-2x)^5 \right) - (25-2x)^4\cdot 30x^2$$ $$=6x \left((25-2x)^5 \right) - 6x \left( (25-2x)^4\cdot 5x \right)$$ $$=6x(25-2x)^4 \cdot(25-2x) - 6x(25-2x)^4 \cdot 5x$$

and now group the terms one more time to get to the answer.