Find (d^(2)t)/(ds^(2)) if t=2s(1-4s)^(2)

Given Function: We are given the function t=2s(1-4s)². To find the second derivative of t with respect to s, we first need to find the first derivative dt/ds.Find First Derivative: Using the product rule for differentiation, which states that (uv)' = u'v + uv', where u = 2s and v = (1-4s)², we differentiate t with respect to s. First, we differentiate u with respect to s: du/ds = 2. Then, we differentiate v with respect to s: dv/ds = d/ds [(1-4s)²] = 2(1-4s)(-4) = -8(1-4s). Now we apply the product rule: dt/ds = du/ds * v + u * dv/ds = 2 * (1-4s)² + 2s * -8(1-4s).Simplify First Derivative: We simplify the expression for the first derivative dt/ds: dt/ds = 2(1-8s+16s²) - 16s(1-4s) = 2 - 16s + 32s² - 16s + 64s².Find Second Derivative: Combine like terms in the expression for dt/ds: dt/ds = 2 + 32s² + 64s² - 32s = 2 + 96s² - 32s.Differentiate Second Derivative: Now we need to find the second derivative d²t/ds² by differentiating dt/ds with respect to s. d²t/ds² = d/ds (2 + 96s² - 32s) = d/ds (2) + d/ds (96s²) - d/ds (32s).Differentiate each term: d/ds (2) = 0, d/ds (96s²) = 192s, and d/ds (32s) = 32. So, d²t/ds² = 0 + 192s - 32.Simplify the expression for the second derivative: d²t/ds² = 192s - 32.

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Algebra 2

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Q. Find

\frac{d^2 t}{ds^2}

t = 2s(1 - 4s)^2

Given Function: We are given the function $t=2s(1-4s)^2$ . To find the second derivative of $t$ with respect to $s$ , we first need to find the first derivative $\frac{dt}{ds}$ .
Find First Derivative: Using the product rule for differentiation, which states that $(uv)' = u'v + uv'$ , where $u = 2s$ and $v = (1-4s)^2$ , we differentiate $t$ with respect to $s$ . First, we differentiate $u$ with respect to $s$ : $\frac{du}{ds} = 2$ . Then, we differentiate $v$ with respect to $s$ : $u = 2s$ $0$ . Now we apply the product rule: $u = 2s$ $1$ .
Simplify First Derivative: We simplify the expression for the first derivative $\frac{dt}{ds}$ : $\frac{dt}{ds} = 2(1-8s+16s^2) - 16s(1-4s) = 2 - 16s + 32s^2 - 16s + 64s^2$ .
Find Second Derivative: Combine like terms in the expression for $\frac{dt}{ds}$ : $\frac{dt}{ds} = 2 + 32s^2 + 64s^2 - 32s = 2 + 96s^2 - 32s$ .
Differentiate Second Derivative: Now we need to find the second derivative $\frac{d^2t}{ds^2}$ by differentiating $\frac{dt}{ds}$ with respect to $s$ . $\frac{d^2t}{ds^2} = \frac{d}{ds} (2 + 96s^2 - 32s) = \frac{d}{ds} (2) + \frac{d}{ds} (96s^2) - \frac{d}{ds} (32s).$
Differentiate Second Derivative: Now we need to find the second derivative $\frac{d^2t}{ds^2}$ by differentiating $\frac{dt}{ds}$ with respect to $s$ . $\frac{d^2t}{ds^2} = \frac{d}{ds} (2 + 96s^2 - 32s) = \frac{d}{ds} (2) + \frac{d}{ds} (96s^2) - \frac{d}{ds} (32s)$ .Differentiate each term: $\frac{d}{ds} (2) = 0$ , $\frac{d}{ds} (96s^2) = 192s$ , and $\frac{d}{ds} (32s) = 32$ . So, $\frac{d^2t}{ds^2} = 0 + 192s - 32$ .
Differentiate Second Derivative: Now we need to find the second derivative $\frac{d^2t}{ds^2}$ by differentiating $\frac{dt}{ds}$ with respect to $s$ . $\frac{d^2t}{ds^2} = \frac{d}{ds} (2 + 96s^2 - 32s) = \frac{d}{ds} (2) + \frac{d}{ds} (96s^2) - \frac{d}{ds} (32s)$ . Differentiate each term: $\frac{d}{ds} (2) = 0$ , $\frac{d}{ds} (96s^2) = 192s$ , and $\frac{d}{ds} (32s) = 32$ . So, $\frac{d^2t}{ds^2} = 0 + 192s - 32$ . Simplify the expression for the second derivative: $\frac{d^2t}{ds^2} = 192s - 32$ .