question let ( k(x)=(4x^{2})g(x)h(x)). given the following table of values, find ( k(7)). provide your…
question let ( k(x)=(4x^{2})g(x)h(x)). given the following table of values, find ( k(7)). provide your answer below: ( k(7)=square) (\begin{array}{|c|c|c|c|c|}hline x&h(x)&g(x)&h(x)&g(x)\\hline4&5& - 6&-5&-2\\hline5&6&1&7&1\\hline7&1&6&1&-1\\hline6&-8&-3&-3&0\\hlineend{array})
Answer
Explanation:
Step1: Apply the product - rule
The product - rule states that if $k(x)=u(x)v(x)$, then $k^{\prime}(x)=u^{\prime}(x)v(x)+u(x)v^{\prime}(x)$. Here, $u(x) = 4x^{2}$ and $v(x)=g(x)h(x)$. First, find the derivative of $u(x)$: $u^{\prime}(x)=\frac{d}{dx}(4x^{2}) = 8x$. Then, by the product - rule $k^{\prime}(x)=8x\cdot g(x)h(x)+4x^{2}\cdot\left[g^{\prime}(x)h(x)+g(x)h^{\prime}(x)\right]$.
Step2: Evaluate at $x = 7$
We need to find the values of $x = 7$ from the table. When $x = 7$, we substitute into $k^{\prime}(x)$. From the table, when $x = 7$, $g(7)=1$, $h(7)=1$, $g^{\prime}(7)= - 1$, $h^{\prime}(7)=1$. And $u^{\prime}(7)=8\times7 = 56$, $u(7)=4\times7^{2}=4\times49 = 196$. [ \begin{align*} k^{\prime}(7)&=8\times7\times g(7)h(7)+4\times7^{2}\times\left[g^{\prime}(7)h(7)+g(7)h^{\prime}(7)\right]\ &=56\times1\times1+196\times\left[(-1)\times1 + 1\times1\right]\ &=56+196\times( - 1 + 1)\ &=56+196\times0\ &=56 \end{align*} ]
Answer:
56