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Use the formula f ′

(x)=lim z→x
z−x
f(z)−f(x)
to find the derivative of the following function. f(x)=4x 2
−5x+1 f ′
(x)=

User Yoanny
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1 Answer

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To find the derivative of the function f(x) = 4x^2 - 5x + 1, we should apply the power rule and addition/subtraction for derivatives. The power rule states that if you have a function of the form f(x) = x^n, then its derivative is given by f'(x) = n*x^(n-1). Moreover, the derivative of a constant is zero.

Step 1:
Let's start by finding the derivative of the first term, 4x^2. With the application of the power rule, its derivative would be 2 * 2 * x^(2-1) = 4x. But remember there's a 4 multiplying x^2, so the derivative becomes 4 * 4x = 16x.

Step 2:
Next, find the derivative of the second term -5x. Using the power rule, its derivative would be 1 * x^(1-1) = x^0 = 1. Multiplying this by the coefficient (-5) gives us -5.

Step 3:
The last term is a constant (1), and the derivative of a constant is zero, so it drops out.

Step 4:
Finally, combine the derivatives of the individual terms to get the derivative of the entire function, f'(x) = 8x - 5.

User Edder
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