Question

the derivative of ( y=(3 x - 7)sqrt{x} ) is ( \frac{dy}{dx}=\frac{bx - 7}{2sqrt{x}} ).
find the value of ( b ).
give an exact answer as an integer.

Explanation:

Step1: Rewrite the function

Rewrite \(y=(3x - 7)\sqrt{x}\) as \(y=(3x - 7)x^{\frac{1}{2}}=3x^{\frac{3}{2}}-7x^{\frac{1}{2}}\).

Step2: Apply the power rule

The power rule is \(\frac{d}{dx}(x^n)=nx^{n - 1}\).
For \(y = 3x^{\frac{3}{2}}-7x^{\frac{1}{2}}\), \(\frac{dy}{dx}=3\times\frac{3}{2}x^{\frac{3}{2}-1}-7\times\frac{1}{2}x^{\frac{1}{2}-1}\).

Step3: Simplify the derivatives

\(\frac{dy}{dx}=\frac{9}{2}x^{\frac{1}{2}}-\frac{7}{2}x^{-\frac{1}{2}}=\frac{9x - 7}{2\sqrt{x}}\)

Answer:

\(9\)