Differentiate 𝑓(𝑥)=4⋅(7−𝑥3).

To differentiate the function 𝑓(𝑥)=4⋅(7−𝑥³), we will use the power rule and the constant multiple rule.

The power rule states that the derivative of 𝑥ⁿ, where n is any real number, is n𝑥ⁿ⁻¹. The constant multiple rule states that the derivative of a constant times a function is the constant times the derivative of the function.

Here are the steps:

1. Identify the outer function and the inner function. In this case, the outer function is 4⋅(u) and the inner function is (7−𝑥³).

2. Differentiate the outer function ignoring the inner function. The derivative of 4⋅(u) with respect to u is 4.

3. Differentiate the inner function. The derivative of (7−𝑥³) with respect to x is 0 - 3𝑥² = -3𝑥² (using the power rule).

4. Multiply the results from steps 2 and 3 according to the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. So, the derivative of 𝑓(𝑥)=4⋅(7−𝑥³) is 4 * -3𝑥² = -12𝑥².