Find the outside and inside functions of the following to find their derivatives:
1) sqrt(2x+9)
2) cos(cos(x))
3) tan(x)
I already know how to find their derivatives I'm just not exactly sure what parts of the chain rule equation would be considered the outside and inside.
The chain rule says that if we have u(x) and f(u(x)),
df/dx = df/du * du/dx
f is the outside function, u is the inside.
So, in the first case
f(u) = √u
u(x) = 2x+9
#2.
f(u) = cos(u)
u(x) = cos(x)
#3. Is almost a trick question.
f(u) = tan(u)
u(x) = x
To find the derivatives using the chain rule, you need to identify the outside and inside functions. Let's break down each function to determine the appropriate parts.
1) The function is sqrt(2x+9). In this case, the outer function is the square root function (sqrt), and the inner function is 2x+9. To differentiate this function, you will treat the square root as the outside function and differentiate it first. Then multiply it by the derivative of the inside function (2x+9). So, the outside function is sqrt(x), and the inside function is 2x+9.
2) The function is cos(cos(x)). Here, the outer function is the cosine function (cos), while the inner function is cos(x). Similar to the first example, you differentiate the outside function first (cos(x)), and then multiply it by the derivative of the inside function (cos'(x)). So, the outside function is cos(x), and the inside function is cos(x).
3) The function is tan(x). In this case, there is no clear separation of an outside and inside function, as tan(x) is its own function. So, you can consider tan(x) as the outer function in this case.
Remember, the chain rule states that the derivative of a composition of functions is given by the derivative of the outside function evaluated at the inside function, multiplied by the derivative of the inside function.