How to Differentiate $x^7(7x+5)^6$

Question

I am trying to differentiate $f(x) = x^7(7x+5)^6$.

So far I have done the following steps:

1) Use the product rule, which is $(x^7(6(7x+5)^5))+((7x^6)(7x+5)^6)$

2) Factor out $x^6$ and $(7x+5)^5$ from the equation, which leaves us with $(x^6(7x+5)^5)(6x+(7(7x+5)))$

3) Simplify $(x^6(7x+5)^5)(6x+(7(7x+5)))$, which gives us $(x^6(7x+5)^5)(55x+35)$

4) I am supposed to simplify the function to the following form: $((x^6)(7x+5)^5)(Ax+B)$, and find what $A$ and $B$ are. So, according to my answer $(x^6(7x+5)^5)(55x+35)$, the answer would be $A=55$ and $B=35$. However, this is not the correct answer...

If anyone knows where I went wrong, please share how you would differentiate this function.

All help is appreciated.

Answer 1

$y=x^7(7x+5)^6$

So $\frac{dy}{dx}=x^7\cdot\{6(7x+5)^5\cdot(7\cdot1+0)\}+(7x+5)^6\cdot\{7x^6\}$

$\frac{dy}{dx}=42x^7(7x+5)^5+7(7x+5)^6x^6$

Answer 2

HINT: by the power and the chain rule we get $$42 (7 x+5)^5 x^7+7 (7 x+5)^6 x^6$$ a possible result is $$7 x^6 (7 x+5)^5 (13 x+5)$$

Answer 3

You went wrong because you didn't apply the chain rule to the second factor. $$[(7x+5)^6]' = 6(7x+5)^5 \cdot[7x+5]'$$

And of course the derivative on the right is just $7$.

Answer 4

f(x)=x^7(7x+5)^6

f(x)=x^7(15,625+131,250x+459,375x^2+857,500x^3+900,375x^4+504,210x^5+117,649x^6)

f(x)=15,625x^7+131,250x^8+45,935x^9+857,500x^10+900,375x^11+504,210x^12+117,649x^13

f'(x)=109,375x^6+1,050,000x^7+413,415x^8+8,575,000x^9+9,904,125x^10+6,050,520x^11+1,529,437x^12