What is the slope of the tangent to the curve `y=4x^3-5x` at ` x=3`.

Options

A. 109
B. 71
C. 103
D. 98

Answer

To find the slope of the tangent to the curve \( y = 4x^3 - 5x \) at \( x = 3 \), 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.

Given:
\[ y = 4x^3 - 5x \]

Differentiating with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(4x^3) - \frac{d}{dx}(5x) \]

Using the power rule \( \frac{d}{dx}(x^n) = nx^{n-1} \):
\[ \frac{dy}{dx} = 12x^2 - 5 \]

### Step 2: Evaluate the derivative at \( x = 3 \)
Now, we need to find the slope of the tangent at the point where \( x = 3 \). We substitute \( x = 3 \) into the derivative we just found.

\[ \frac{dy}{dx} \bigg|_{x=3} = 12(3^2) - 5 \]

Calculating \( 3^2 \):
\[ 3^2 = 9 \]

Now substituting back:
\[ \frac{dy}{dx} \bigg|_{x=3} = 12(9) - 5 \]

Calculating \( 12 \times 9 \):
\[ 12 \times 9 = 108 \]

So,
\[ \frac{dy}{dx} \bigg|_{x=3} = 108 - 5 = 103 \]

### Conclusion
The slope of the tangent to the curve at \( x = 3 \) is:
\[ \text{slope} = 103 \]

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