Questions: If f(x)=6x^2-7x-40, find f'(x).

Transcript text: If $f(x)=6 x^{2}-7 x-40$, find $f^{\prime}(x)$.

Solution

Solution Steps

Step 1: Identify the function

The given function is $ f(x) = 6x^{2} - 7x - 40 $.

Step 2: Apply the power rule

To find the derivative $ f^{\prime}(x) $, apply the power rule to each term. The power rule states that if $ f(x) = ax^{n} $, then $ f^{\prime}(x) = n \cdot a \cdot x^{n-1} $.

Step 3: Differentiate each term

For the term $ 6x^{2} $: \[ \frac{d}{dx}(6x^{2}) = 2 \cdot 6 \cdot x^{2-1} = 12x \]
For the term $ -7x $: \[ \frac{d}{dx}(-7x) = 1 \cdot (-7) \cdot x^{1-1} = -7 \]
For the constant term $ -40 $: \[ \frac{d}{dx}(-40) = 0 \]

Step 4: Combine the derivatives

Combine the derivatives of each term to find $ f^{\prime}(x) $: \[ f^{\prime}(x) = 12x - 7 \]