Derivative of (x^2 + 2)(x - 5)

To find the derivative of the given function (x^2 + 2)(x - 5), we will use the product rule of differentiation. The product rule states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.

Applying the product rule:
Let f(x) = x^2 + 2 and g(x) = x - 5

Now, f'(x) = 2x and g'(x) = 1

Using the product rule,
(dy/dx) = f'(x)g(x) + f(x)g'(x)
(dy/dx) = (2x)(x - 5) + (x^2 + 2)(1)
(dy/dx) = 2x^2 - 10x + x^2 + 2
(dy/dx) = 3x^2 - 10x + 2

Therefore, the derivative of (x^2 + 2)(x - 5) is 3x^2 - 10x + 2.

Derivative of 8^x * log(x)

To find the derivative of the function 8^x * log(x), we will use the product rule as well as the chain rule of differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

Applying the product rule and chain rule:
Let f(x) = 8^x and g(x) = log(x)

Now, f'(x) = 8^x * log(8) and g'(x) = 1/x

Using the product rule and chain rule,
(dy/dx) = f'(x)g(x) + f(x)g'(x)
(dy/dx) = (8^x * log(8)) * log(x) + 8^x * (1/x)
(dy/dx) = 8^x * log(8) * log(x) + 8^x / x

Therefore, the derivative of 8^x * log(x) is 8^x * log(8) * log(x) + 8^x / x.