Find slope of tangent if equation of the curve is `y= 3 x^4-4x` at `x=4`

Answer

To find the slope of the tangent to the curve given by the equation \( y = 3x^4 - 4x \) at the point where \( x = 4 \), we will follow these steps:

### Step 1: Differentiate the function
We need to find the derivative of the function \( y \) with respect to \( x \). The derivative \( \frac{dy}{dx} \) gives us the slope of the tangent line at any point on the curve.

The function is:
\[ y = 3x^4 - 4x \]

Using the power rule for differentiation, which states that \( \frac{d}{dx}(x^n) = nx^{n-1} \), we differentiate each term:

1. The derivative of \( 3x^4 \) is \( 12x^3 \) (since \( 4 \cdot 3 = 12 \) and the power decreases by 1).
2. The derivative of \( -4x \) is \( -4 \) (since the derivative of \( x \) is 1).

Thus, the derivative is:
\[ \frac{dy}{dx} = 12x^3 - 4 \]

### Step 2: Evaluate the derivative at \( x = 4 \)
Next, we substitute \( x = 4 \) into the derivative to find the slope of the tangent at that point.

\[ \frac{dy}{dx} \bigg|_{x=4} = 12(4^3) - 4 \]

Calculating \( 4^3 \):
\[ 4^3 = 64 \]

Now substituting this value back into the derivative:
\[ \frac{dy}{dx} \bigg|_{x=4} = 12 \cdot 64 - 4 \]

Calculating \( 12 \cdot 64 \):
\[ 12 \cdot 64 = 768 \]

Now, subtracting 4:
\[ 768 - 4 = 764 \]

### Conclusion
The slope of the tangent to the curve at \( x = 4 \) is:
\[ \text{slope} = 764 \]