Question

Given the function f(x) = 4x3 + 13x, determine the gradient of the function when x = 2.

Answer 1

Answer:

The gradient for the function 4x³ + 13x when x is 2 is 61.

Step-by-step explanation:

Before we do anything, let's find the partial derivative of 4x³ + 13x.

d/dx (4x³ + 13x)

Step 1:The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of axⁿ  is naxⁿ⁻¹.

3 * 4x³⁻¹ + 13x¹⁻¹

Step 2: Multiply 3 and 4.

12x³⁻¹ + 13x¹⁻¹

Steps 3 & 4: Subtract 1 from 3 (Step 3) and 1 (Step 4) respectively.

12x² + 13

Note: 13x⁰ simply becomes 13 because t⁰ = t.

Now that we know the derivative, substitute x for 2.

12(2²) + 13

Step 5: Square 2 to get 4.

12(4) + 13

Step 6: Multiply 12 and 4 to get 48.

48 + 13

Step 7: Add 48 and 13 to get 61.

61

So, the gradient of 4x³ + 13x is 61, if x is 2.