Derivative of f(x) = (2ax + b)c + 2x² + 1

To find the derivative of the given function f(x), we need to apply the rules of differentiation. Let's break down the function into its individual terms and apply the rules step by step.

Step 1: Differentiating (2ax + b)c
The derivative of c with respect to x is 0, as c is a constant. So, we can ignore the derivative of c in this step.

To differentiate (2ax + b), we use the power rule for differentiation. The power rule states that the derivative of x^n is equal to n*x^(n-1), where n is a constant.

In this case, the derivative of (2ax + b) with respect to x is:
2a * d/dx(x) + 0 = 2a

Therefore, the first term (2ax + b)c differentiates to 2a.

Step 2: Differentiating 2x² + 1
To differentiate 2x², we use the power rule. The derivative of x^2 is 2x.

Therefore, the second term 2x² differentiates to 4x.

The derivative of the constant term 1 is 0.

Step 3: Combining the derivatives
Now that we have the derivatives of each term, we can combine them to find the derivative of the entire function f(x).

f'(x) = derivative of (2ax + b)c + derivative of 2x² + derivative of 1
= 2a + 4x + 0
= 2a + 4x

Explanation:
The derivative of f(x) is denoted as f'(x) and represents the rate of change of the function with respect to x. It gives us the slope of the function at any given point.

In this case, the function f(x) is a combination of two terms: (2ax + b)c and 2x² + 1. We differentiate each term separately using the rules of differentiation.

The derivative of (2ax + b)c is 2a, as c is a constant and its derivative is zero. The derivative of 2x² is 4x, using the power rule. The derivative of the constant term 1 is 0.

By combining these derivatives, we obtain the derivative of the entire function f(x) as f'(x) = 2a + 4x.

Conclusion:
The derivative of the given function f(x) = (2ax + b)c + 2x² + 1 is f'(x) = 2a + 4x. The derivative represents the rate of change of the function with respect to x, and it gives us the slope of the function at any given point.

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