Calculating Derivative of a Function and Finding its Value
The image shows a function f(x) = 2x^7 - 5x + 1 and asks to calculate f'(x) given f(x) = 1.
To find f'(x), we need to differentiate f(x) with respect to x. Taking the derivative term by term, we get:
f'(x) = d/dx (2x^7) - d/dx (5x) + d/dx (1)
Using the power rule for differentiation, which states that d/dx (x^n) = n*x^(n-1), we get:
f'(x) = 7 * 2x^(7-1) - 5 * 1x^(1-1)
f'(x) = 14x^6 - 5
Now we have an expression for f'(x). However, the problem states that this should be evaluated given f(x) = 1. This means we have to find x such that 2x^7 - 5x + 1 = 1. From the equation, we can simplify to:
2x^7 - 5x = 0
Removing the constant term from both sides since it does not impact finding x for which f(x) = 1, we need an additional step to find x that satisfies this equation. Generally, this would require solving a 7th-degree polynomial, which may or may not be solvable in terms of radicals.
However, the task does not specify needing to find the value of x that makes f(x) = 1. Instead, if you need to find f'(1), then you would substitute x with 1:
f'(1) = 14(1)^6 - 5
f'(1) = 14 - 5
f'(1) = 9
So, if you're required to find f'(1), the answer is 9. If you have to find f'(x) for the value of x such that f(x) = 1, the determination of x from the given equation 2x^7 - 5x + 1 = 1 will be more complex and may not have a straightforward solution.
