Question
Find the derivative of the function f(x) = 2x^2 + 3x - 5 using the product rule.
Answers
GPT-4o mini
To find the derivative of the function \( f(x) = 2x^2 + 3x - 5 \) using the product rule, we first need to express the function in a form suitable for applying the product rule. The product rule states that if you have a function \( u(x)v(x) \), then the derivative \( \left( u v \right)' = u'v + uv' \).
However, the function \( f(x) \) is not immediately in product form. But we can treat it as the product of two functions, for example \( u(x) = 2x^2 \) and \( v(x) = 1 + \frac{3x}{2x^2} - \frac{5}{2x^2} \), though in this case it is a bit convoluted for \( f \).
Instead, it's simpler and more appropriate to just calculate the derivative directly using the power rule:
1. The derivative of \( 2x^2 \) is \( 4x \).
2. The derivative of \( 3x \) is \( 3 \).
3. The derivative of the constant \(-5\) is \( 0 \).
Now, we sum these derivatives:
\[
f'(x) = 4x + 3.
\]
In conclusion, since the product rule isn’t directly applicable here, the derivative of \( f(x) = 2x^2 + 3x - 5 \) is:
\[
f'(x) = 4x + 3.
\]
However, the function \( f(x) \) is not immediately in product form. But we can treat it as the product of two functions, for example \( u(x) = 2x^2 \) and \( v(x) = 1 + \frac{3x}{2x^2} - \frac{5}{2x^2} \), though in this case it is a bit convoluted for \( f \).
Instead, it's simpler and more appropriate to just calculate the derivative directly using the power rule:
1. The derivative of \( 2x^2 \) is \( 4x \).
2. The derivative of \( 3x \) is \( 3 \).
3. The derivative of the constant \(-5\) is \( 0 \).
Now, we sum these derivatives:
\[
f'(x) = 4x + 3.
\]
In conclusion, since the product rule isn’t directly applicable here, the derivative of \( f(x) = 2x^2 + 3x - 5 \) is:
\[
f'(x) = 4x + 3.
\]