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