Question

Answer the Following problem about derivatives.

Answer 1

The derivatives for the given functions are as follows:

a) -3.

b) -1.

c) 1.

d) 0.

What is the product rule for a derivative?

The product rule for a derivative is given as follows:

[f(x)g(x)]' = f'(x)g(x) + g'(x)f(x).

Hence, at x = 6, we have that:

[f(x)g(x)]'(6) = f'(6)g(6) + g'(6)f(6).

Replacing the values given in this problem, we have that the answer for item a is:

[f(x)g(x)]'(6) = f'(6)g(6) + g'(6)f(6) = 2(-1) - 1(1) = -2 - 1 = -3.

What is the quotient rule for a derivative?

The quotient rule for a derivative is given as follows:

$\left(\frac{f(x)}{g(x)}\right)^{\prime} = \frac{f^{\prime}(x)g(x) - g^{\prime}(x)f(x)}{g(x)^2}$

Hence, at x = 6, we have that:

$\left(\frac{f(x)}{g(x)}\right)^{\prime}(6) = \frac{f^{\prime}(6)g(6) - g^{\prime}(6)f(6)}{g(6)^2}$

Then the derivative in item b is:

[2(-1) - (-1)(1)]/[(-1)^2] = -1/1 = -1.

What is the derivative for the square root of a function?

Applying the chain rule, the derivative is given by:

$(\sqrt{f(x)})^{\prime} = \frac{1}{2\sqrt{f(x)}}f^{\prime}(x)$

Replacing at x = 6, the derivative for item c is given by:

1/2 x 2 = 1.

What is the derivative of a constant?

The derivative of a constant is of 0. In item d, the multiplication of f(6) by g'(6) results in a constant, hence the derivative is of 0.

More can be learned about derivative rules at brainly.com/question/25081524

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