SOLUTION: Find the derivative of the function. y=-8xln(5x+7) dy/dx= ? Please answer the dy/dx. Thank you

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Question 1191921: Find the derivative of the function.
y=-8xln(5x+7)
dy/dx= ?
Please answer the dy/dx. Thank you
Found 2 solutions by ikleyn, MathLover1:
Answer by ikleyn(52771) About Me  (Show Source):
You can put this solution on YOUR website!
.

Just solved under this link

https://www.algebra.com/algebra/homework/Finance/Finance.faq.question.1191920.html



Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

y=-8x%2Aln%285x%2B7%29
dy%2Fdx= ?
Differentiate separately both sides of the equation (treat y as a function of x):

dy%2Fdx=+%28d%2Fdx%29%28+-8x%2Aln%285x%2B7%29%29
Differentiate the RHS of the equation. Apply the constant multiple rule.

%28d%2Fdx%29%28+-8x%2Aln%285x%2B7%29%29

=-8%28d%2Fdx%29%28+x%2Aln%285x%2B7%29%29...........Apply the product rule

=-8%28%28d%2Fdx%29%28+x%29%2Aln%285x%2B7%29%2Bx%28d%2Fdx%29%28ln%285x%2B7%29%29%29

=-8%28d%2Fdx%29%28+x%29%2Aln%285x%2B7%29-8%2Ax%28d%2Fdx%29%28ln%285x%2B7%29%29

The function ln%285x%2B7%29 is the composition f%28g%28x%29%29+of two functions f%28u%29=ln%28u%29+and g%28x%29=5x%2B7, so

=......The derivative of the natural logarithm is +%28d%2Fdu%29%28ln%28u%29%29=+1%2Fu
​
=-8x%28+1%2Fu%29+%28d%2Fdx%29%285x%2B7%29-8ln%285x%2B7%29+%28d%2Fdx%29%28x%29
=%28-+8x%28d%2Fdx%29%285x%2B7%29%29%2Fu-8ln%285x%2B7%29+%28d%2Fdx%29%28x%29

Return to the old variable:

=%28-+8x%28d%2Fdx%29%285x%2B7%29%29%2F%285x%2B7%29-8ln%285x%2B7%29+%28d%2Fdx%29%28x%29

The derivative of a sum/difference is the sum/difference of derivatives:

=

The derivative of a constant is 0:%28d%2Fdx%29%287%29=0

=%28-+8x%28d%2Fdx%29%285x%29%2B0%29%29%2F%285x%2B7%29-8ln%285x%2B7%29+%28d%2Fdx%29%28x%29

Apply the constant multiple rule:%28d%2Fdx%29%285x%29=5%28d%2Fdx%29%28x%29=5%2A1=5

=%28-+8x%285%29%29%2F%285x%2B7%29-8ln%285x%2B7%29%2A1

=%28-+40x%29%2F%285x%2B7%29-8ln%285x%2B7%29+

= -%2840x%2B8%285x%2B7%29ln%285x%2B7%29%29%2F%285x%2B7%29

Thus,

Therefore, %28dy%2Fdx%29=-%2840x%2B8%285x%2B7%29ln%285x%2B7%29%29%2F%285x%2B7%29
​