This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
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To differentiate $y = 5x^2 + 2x + 5$, we will use the power rule for differentiation, which states that $\frac{d}{dx}(ax^n) = nax^{n-1}$, and the rule that the derivative of a constant is zero.
Step 1: Differentiate each term with respect to $x$.
For the first term, $5x^2$:
$$ \frac{d}{dx}(5x^2) = 2 \cdot 5x^{2-1} = 10x^1 = 10x $$
For the second term, $2x$:
$$ \frac{d}{dx}(2x) = 1 \cdot 2x^{1-1} = 2x^0 = 2 \cdot 1 = 2 $$
For the third term, $5$ (a constant):
$$ \frac{d}{dx}(5) = 0 $$
Step 2: Combine the derivatives of each term.
$$ \frac{dy}{dx} = 10x + 2 + 0 $$
$$ \frac{dy}{dx} = 10x + 2 $$
The derivative of $y = 5x^2 + 2x + 5$ is:
$$ \boxed{\frac{dy}{dx} = 10x + 2} $$
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