Final answer:
To find the second derivative of the function w=10z²e˞z, we apply the product rule and chain rule to first find the first derivative dw/dz and then differentiate again to find d²w/dz². The final second derivative of the function is 10(2e˞z + 4ze˞z + z²e˞z).
Step-by-step explanation:
To find the second derivative of the function w = 10z²ez with respect to z, we need to apply the product rule and the chain rule of differentiation. The product rule states that the derivative of the product of two functions is the first function multiplied by the derivative of the second function plus the second function multiplied by the derivative of the first function. The chain rule tells us how to differentiate composite functions.
Applying these rules, we first find the first derivative dw/dz and then differentiate it again to find d²w/dz².
The first derivative is:
dw/dz = d(10z²ez)/dz = 10 d(z²ez)/dz = 10(2zez + z²ez)
Now, we find the second derivative:
d²w/dz² = d(10(2zez + z²ez))/dz = 10(2ez + 2zez + 2zez + z²ez) = 10(2ez + 4zez + z²ez)