Question

find the gradient of the graph of ( y=(2 - 3x)(1 - x) ) at the point, where ( x = 7 ).

Explanation:

Step1: Expand the function

$$\begin{align*} y&=(2 - 3x)(1 - x)\\ &=2\times(1 - x)-3x\times(1 - x)\\ &=2-2x-3x + 3x^{2}\\ &=3x^{2}-5x + 2 \end{align*}$$

Step2: Find the derivative

The derivative of \(y = ax^{n}\) is \(y^\prime=anx^{n - 1}\). For \(y = 3x^{2}-5x + 2\), we have:
\(y^\prime=\frac{d}{dx}(3x^{2})-\frac{d}{dx}(5x)+\frac{d}{dx}(2)\)
\(y^\prime=3\times2x-5\times1+0 = 6x-5\)

Step3: Substitute \(x = 7\) into the derivative

When \(x = 7\), \(y^\prime=6\times7-5\)
\(y^\prime=42 - 5=37\)

Answer:

\(37\)