Scelta multipla
Find the third derivative of the given function.
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3. Techniques of Differentiation
Higher Order Derivatives
Scelta multipla
Find the third derivative of the given function.
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Start by identifying the given function: \( y = 3x^2 + 9x + 1 \). The goal is to find the third derivative, which means we will differentiate the function three times.
First, compute the first derivative \( y' \) by applying the power rule \( \frac{d}{dx}[x^n] = n \cdot x^{n-1} \): \( y' = \frac{d}{dx}[3x^2] + \frac{d}{dx}[9x] + \frac{d}{dx}[1] \).
Simplify the first derivative: \( y' = 6x + 9 \), since the derivative of \( 3x^2 \) is \( 6x \), the derivative of \( 9x \) is \( 9 \), and the derivative of a constant \( 1 \) is \( 0 \).
Next, compute the second derivative \( y'' \) by differentiating \( y' \): \( y'' = \frac{d}{dx}[6x] + \frac{d}{dx}[9] \).
Simplify the second derivative: \( y'' = 6 \), since the derivative of \( 6x \) is \( 6 \) and the derivative of a constant \( 9 \) is \( 0 \). Finally, compute the third derivative \( y''' \): \( y''' = \frac{d}{dx}[6] \), which simplifies to \( 0 \), as the derivative of a constant is always \( 0 \).
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