If $ f(x) = x^2 + 3x $, then $ f'(x) $ is:

Question

If the equation $ \sin^4 x - (p + 2)\sin^2 x - (p + 3) = 0 $ has a solution, then $ p $ must lie in the interval:

Answer 1

To determine the derivative of the function $ f(x) = x^2 + 3x $, the power rule of differentiation is applied. The power rule is stated as:

$\frac{d}{dx}[x^n] = nx^{n-1}$

Applying the power rule to each component of $ f(x) $:

1. For the term $x^2$:
$\frac{d}{dx}[x^2] = 2x^{2-1} = 2x$

2. For the term $3x$:
$\frac{d}{dx}[3x] = 3$

Consequently, the derivative $ f'(x) $ is found by differentiating each term and summing the results:

$f'(x) = 2x + 3$

Therefore, the derivative is $2x + 3$.

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