Question

If f(x) = 9(sin(x))x, find f'(1).

Answer 1

To find f'(1) for the function f(x) = 9(sin(x))x, we can use the product rule. After substituting x = 1, we find that f'(1) is equal to 7.029.

Step-by-step explanation:

To find f'(1) for the function f(x) = 9(sin(x))x, we can use the product rule. The product rule states that the derivative of a 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.

Using the product rule, we have:

f'(x) = 9(sin(x))x' + x(9(sin(x))'.

The derivative of sin(x) is cos(x), so f'(x) = 9(sin(x)) + x(9cos(x)).

Now, we can substitute x = 1 to find f'(1):

f'(1) = 9(sin(1)) + 1(9cos(1)).

After calculating the values for sin(1) and cos(1), we get f'(1) = 9(0.841) + 1(-0.540) = 7.569 - 0.540 = 7.029.

Answer 2

The calculation of f'(1) for f(x) = 9(sin(x))^x entails the application of the product, chain, and power rules of differentiation, but the correct functional form must be established.

Step-by-step explanation:

To calculate f'(1) for the function f(x) = 9(sin(x))x, we can use the chain rule and the power rule of differentiation. However, the expression provided appears to contain a typo or incorrect formulation because (sin(x))x is an unusual function to differentiate at x=1 and would not typically be represented in a manner suggesting that sin(x) equals 1 for all forces, as forces are not involved in this mathematical function itself. Assuming that we proceed with the differentiation of the given function as is:

To find the derivative of f(x), we apply the product rule to the function of a function. The derivative of f(x) with respect to x when x = 1 is:

f'(x) = d/dx [9(sin(x))x]

Since this is a product of two functions, 9 and (sin(x))x, and (sin(x))x is itself a function raised to a power, we use the chain rule and power rule:

f'(x) = 9 * d/dx [(sin(x))x]

f'(x) = 9 * x(sin(x))x-1 * cos(x) + 9 * ln(sin(x)) * (sin(x))x

The actual calculation of f'(1) would need the correct functional form of f(x) and could involve a numerical value or an application of numerical integration techniques if the function is not easily integrable analytically.