Question

For f(x) = √(3x + 28), find f'(x) using the definition of the derivative.

Find the second derivative for y = sin(x) + e^(5x).

Answer 1

The derivative of f(x) = √(3x + 28) is 3/(2√(3x + 28)), and the second derivative of y = sin(x) + e^(5x) is -sin(x) + 25e^(5x).

Step-by-step explanation:

To find the derivative of f(x) = √(3x + 28), we use the chain rule. The derivative of the outer function, √u (where u = 3x + 28), is 1/(2√u). The derivative of the inner function, u with respect to x, is 3. Therefore, the derivative of f with respect to x is:

f'(x) = 1/(2√(3x + 28)) * 3 = 3/(2√(3x + 28)).

For the second derivative of y = sin(x) + e^(5x), we take the first derivative, which is cos(x) + 5e^(5x), and differentiate again:

For the sin(x) term, the second derivative will be -sin(x), as the derivative of cos(x) is -sin(x).

For the e^(5x) term, applying the chain rule, the derivative of 5e^(5x) with respect to x is 25e^(5x).

Thus, the second derivative is y'' = -sin(x) + 25e^(5x).