Question

If f is a differentiable function and y=sin(f(x2)) what is dydx when x = 3 ?

Answer 1

The derivative dy/dx of the function y=sin(f(x^2)) when x = 3 is calculated using the chain rule, resulting in cos(f(9)) • 6 • f'(9). The values of f(9) and f'(9) are required to find the exact numerical value.

Step-by-step explanation:

The student is asking for the derivative of the function y=sin(f(x^2)) with respect to x when x = 3. To find dy/dx, we will use the chain rule and the fact that f is differentiable. The chain rule tells us that the derivative of sin(g(x)) with respect to x is cos(g(x)) • g'(x). Applying this to our function, we first let g(x) = f(x^2). The derivative of g with respect to x, g'(x), is 2x • f'(x^2). Therefore, the derivative of y with respect to x, dy/dx, is cos(f(x^2)) • 2x • f'(x^2).

At x = 3, dy/dx becomes cos(f(9)) • 2 • 3 • f'(9). To complete this, we need the values of f(9) and f'(9), which are not provided. Therefore, the answer in terms of f and its derivative is cos(f(9)) • 6 • f'(9).

Answer 2

To calculate dy/dx when y = sin(f(x^2)), the chain rule must be applied. The derivative at x = 3 depends on the function f and its derivative at 9; it is cos(f(9)) · 6 · f'(9). The exact value requires more information about f.

Step-by-step explanation:

To find dy/dx when f is a differentiable function and y = sin(f(x2)), we need to use the chain rule. First, we differentiate sin(f(u)) with respect to u, and then we differentiate u = f(x2) with respect to x. The chain rule states that if y = g(h(x)), then dy/dx = (dg/dh)(dh/dx). Here, g(u) = sin(u) and h(x) = f(x2), so:

dy/du = cos(u)

du/dx = 2x · f'(x2)

Thus, dy/dx is the product of these two derivatives. When x = 3, substitute u with f(32) into dy/du, and substitute x with 3 into du/dx, and then multiply them together:

dy/dx at x = 3 = cos(f(9)) · 2 · 3 · f'(9)

Note that the exact value of dy/dx at x = 3 cannot be determined without the specific form of the function f.