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 will follow these steps:

### Step 1: Differentiate the function
We start by differentiating the function \( y = x^4 - 4x \) with respect to \( x \).

\[ \frac{dy}{dx} = \frac{d}{dx}(x^4) - \frac{d}{dx}(4x) \]

### Step 2: Apply the power rule
Using the power rule for differentiation, we find:

\[ \frac{dy}{dx} = 4x^3 - 4 \]

### Step 3: Substitute \( x = 4 \)
Now, we substitute \( x = 4 \) into the derivative to find the slope of the tangent at that point.

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

### Step 4: Calculate \( 4^3 \)
Calculate \( 4^3 \):

\[ 4^3 = 64 \]

### Step 5: Substitute back into the derivative
Now substitute \( 64 \) back into the derivative:

\[ \frac{dy}{dx} \bigg|_{x=4} = 4(64) - 4 \]

### Step 6: Simplify the expression
Now, simplify the expression:

\[ \frac{dy}{dx} \bigg|_{x=4} = 256 - 4 = 252 \]

### Conclusion
Thus, the slope of the tangent to the curve \( y = x^4 - 4x \) at \( x = 4 \) is \( 252 \).