How do you differentiate #y=(5x^5+5)(-2x^5-3)# using the product rule?

Question

Ezra Adams · Accepted Answer

#dy/dx=-100x^9-125x^4# #"Given " y=g(x).h(x)" then"# #color(red)(bar(ul(|color(white)(2/2)color(black)(dy/dx=g(x)h'(x)+h(x)g'(x))color(white)(2/2)|)))larr" product rule"# #"here " g(x)=5x^5+5rArrg'(x)=25x^4# #"and " h(x)=-2x^5-3rArrh'(x)=-10x^4# #rArrdy/dx=(5x^5+5)(-10x^4)+(-2x^5-3)(25x^4)# #color(white)(rArrdy/dx)=-50x^9-50x^4-50x^9-75x^4# #color(white)(rArrdy/dx)=-100x^9-125x^4#

Isaiah Allen · Answer

undefined To differentiate y=(5x^5+5)(-2x^5-3) using the product rule, follow these steps:

1. Identify the two functions being multiplied: $ u = 5x^5 + 5 $ and $ v = -2x^5 - 3 $.

2. Apply the product rule, which states that the derivative of the product of two functions $ u $ and $ v $ is given by $ u'v + uv' $.

3. Find the derivatives of $ u $ and $ v $: $ u' = 25x^4 $ and $ v' = -10x^4 $.

4. Substitute these values into the product rule formula: $ y' = (25x^4)(-2x^5 - 3) + (5x^5 + 5)(-10x^4) $.

5. Simplify the expression: $ y' = -50x^9 - 75x^4 + (-50x^9 - 50x^4) $.

6. Combine like terms: $ y' = -100x^9 - 125x^4 $.

Therefore, the derivative of $ y = (5x^5 + 5)(-2x^5 - 3) $ using the product rule is $ y' = -100x^9 - 125x^4 $.