Question 1113571: CALCULUS. Find (d^2 y)/(dx^2) if y=(x^2 +1)^5. Please help. Thankyou in advance! Found 2 solutions by math_helper, Alan3354:Answer by math_helper(2461) (Show Source):
You can put this solution on YOUR website!
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For the first part we can use u-substitution:
Let —>
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Aside:
Notice that the chain rule (which applies to composite functions) could have been used:
The chain rule says (f(g(x)))' = f'(g)*g') "The derivative of f of g(x) is the derivative of f(g(x)) times the derivative of g(x)" Often used for expressions like (mx+k)^n where, m and k are numbers, g(x)=mx+k and f(x)=x^n, one takes n*(mx+k)^(n-1) then multiplies by dg/dx = d(mx+k)/dx = m to arrive at m(mx+k)^(n-1). Think of it as an implicit use of u-substitution.
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I mention the chain rule because I will use it for finding the 2nd derivative. However, that step happens to be identical in procedure to how we found the first derivative.
Now onward to get …
— is a product of two functions 10x and (x^2+1)^4
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The product of functions lends itself to using the product rule:
The product rule is (f*g)' = fg' + f'g (where f=f(x) and f' = df/dx, g=g(x), g'=dg/dx)
"The first times the derivative of the 2nd, plus the 2nd times the derivative of the first."
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The first is "10x" , the 2nd is "(x^2+1)^4" so we will do 10x * derivative of(x^2+1)^4) + (x^2+1)^4 * derivative of(10x), noting that finding the derivative of (x^2+1)^4 wrt x is very similar to how we found the first derivative (it is the same process just different numbers). The part I'm referring to is highlighted in below.
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Which can be re-written after much algebra:
and that factors (thanks to an online factoring tool) to:
