Lara T. answered • 07/31/23
Applied Math Tutor Majoring in Physics, Astrophysics, & Math @ UF
We can use chain rule to find the second derivative of this function. Let's start with the first derivative.
We can see that the inner function here is 2-7x, while the outer function is x4. The formula for chain rule is f'(g(x))⋅g'(x), where f(x) is the outer function and g(x) is the inner function. Taking the derivative of our outer function, x4, gives us 4x3, using power rule. f'(g(x)) in the chain rule formula tells us that we must compose 4x3 and 2-7x, giving us 4(2-7x)3. Now, we must multiply this by the derivative of 2-7x, which would be -7. Therefore, we have -28(2-7x)3.
However, we must not forget to multiply by the 5 which was present at the beginning of the original function. This gives us the first derivative of -140(2-7x)3.
Now, we may proceed with the second derivative. We will follow the same steps. The inner function is 2-7x, and the outer function is x3. The derivative of the outer function is 3x2, and the derivative of the inner function is -7. We can plug this into the chain rule formula by first composing f'(x) and g(x), giving us 3(2-7x)2. Then, we multiply that by the derivative of the inner function, which gives us -21(2-7x)2.
Finally, we multiply by the -140 present in the first derivative, giving us 2,940(2-7x)2 as our second derivative. Hope this helps!
