How do you find the derivative of this function:
G(x) = (x^4 - x +1)^2 * (x^2 - 2)^3
d/dx [f(x)g(x)= f d/dx g(x)+ d/dx [f(x)] 2(x^2-2)^2 (3(x^4-x+1) x+(x^2-2) (4x^3-1)) (x^4-x+1)
y = (x^4 - x +1)^2 * (x^2 - 2)^3
Using the product rule and chain rule,
y' = 2(x^4-x+1)(4x^3-1)(x^2-2)^3 + (x^4-x+1)^2*3(x^2-2)^2 (2x)
= (x^4-x+1)(x^2-2)^2 (2(4x^3-1)(x^2-2)+(x^4-x+1)*3*2x)
= x^14 - 6x^12 - 2x^11 + 14x^10 + 12x^9 - 19x^8 - 26x^7 + 19x^6 + 28x^5 - 10x^4 - 24x^3 + 4x^2 + 16x - 8
Homework helper's expression looks good, but yields
14x^13 - 72x^11 - 22x^10 + 140x^9 + 108x^8 - 152x^7 - 182x^6 + 114x^5 140x^4 - 40x^3 - 72x^2 - 8x + 16
Can you spot the error?
To find the derivative of a function, we can use the chain rule and the power rule of differentiation. The chain rule states that if we have a composite function, we need to differentiate the outer function, and then multiply it by the derivative of the inner function. The power rule states that if we have a function raised to a power, we can bring down the exponent as a coefficient, and then subtract one from the exponent.
Let's break down the given function G(x) = (x^4 - x + 1)^2 * (x^2 - 2)^3 into two parts: f(x) = (x^4 - x + 1)^2 and g(x) = (x^2 - 2)^3.
First, let's find the derivative of f(x) using the chain rule and the power rule. We start by differentiating the outer function, which is squaring. The derivative of the outer function is 2 times the function to the power of (2 - 1):
f'(x) = 2 * (x^4 - x + 1)^(2 - 1) * (derivative of x^4 - x + 1)
Next, we need to find the derivative of the inner function, x^4 - x + 1. We can do this by applying the power rule to each term individually:
The derivative of x^4 = 4x^(4 - 1) = 4x^3
The derivative of -x = -1
The derivative of 1 = 0 since it's a constant
So, the derivative of the inner function is:
f'(x) = 2 * (x^4 - x + 1) * (4x^3 - 1)
Now, let's find the derivative of g(x) using the chain rule and the power rule. Again, we start by differentiating the outer function, which is raising to the power of 3. The derivative of the outer function is 3 times the function to the power of (3 - 1):
g'(x) = 3 * (x^2 - 2)^(3 - 1) * (derivative of x^2 - 2)
Next, we find the derivative of the inner function x^2 - 2:
The derivative of x^2 = 2x
The derivative of -2 = 0
So, the derivative of the inner function is:
g'(x) = 3 * (x^2 - 2) * 2x
Now, we can find the final derivative of G(x) by multiplying the derivatives of f(x) and g(x):
G'(x) = f'(x) * g(x) + f(x) * g'(x)
Therefore,
G'(x) = (2 * (x^4 - x + 1) * (4x^3 - 1)) * (x^2 - 2)^3 + (x^4 - x + 1)^2 * (3 * (x^2 - 2) * 2x)
Simplifying this expression will give you the derivative of G(x).