The slope of the tangent to the curve `y=x^4-4x` at x=4 is:

Options

A. 100
B. 200
C. 50
D. 252

Answer

To find the slope of the tangent to the curve \( y = x^4 - 4x \) at \( x = 4 \), we need to follow these steps:

### Step 1: Differentiate the function
We start by finding 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:
\[ y = x^4 - 4x \]

Differentiating with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(x^4) - \frac{d}{dx}(4x) \]
Using the power rule:
\[ \frac{dy}{dx} = 4x^3 - 4 \]

### Step 2: Evaluate the derivative at \( x = 4 \)
Now, we need to find the slope of the tangent at \( x = 4 \) by substituting \( x = 4 \) into the derivative we found.

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

Calculating \( 4^3 \):
\[ 4^3 = 64 \]
Now substituting back:
\[ \frac{dy}{dx} \bigg|_{x=4} = 4(64) - 4 \]
Calculating \( 4(64) \):
\[ 4(64) = 256 \]
So,
\[ \frac{dy}{dx} \bigg|_{x=4} = 256 - 4 = 252 \]

### Conclusion
The slope of the tangent to the curve \( y = x^4 - 4x \) at \( x = 4 \) is:
\[ \boxed{252} \]