Differentiate the function.
f(x) = sin(7 ln x)
7/x (cos(7lnx))
Why did the sine function go to the therapist? Because it couldn't find its period!
To differentiate the given function f(x) = sin(7 ln x), we can use the chain rule.
Let's break it down step-by-step:
Step 1: Identify the inner function.
In this case, the inner function is 7 ln x.
Step 2: Calculate the derivative of the inner function.
The derivative of ln x is 1/x.
So, the derivative of 7 ln x is 7/x.
Step 3: Differentiate the outer function using the chain rule.
The derivative of sin u, where u is a function of x, is cos u times the derivative of u.
Therefore, the derivative of sin(7 ln x) is cos(7 ln x) times the derivative of 7 ln x.
Step 4: Multiply the derivatives calculated in steps 2 and 3.
The derivative of 7 ln x is 7/x, and the derivative of sin(7 ln x) is cos(7 ln x).
So, the overall derivative of the function f(x) = sin(7 ln x) is:
f'(x) = (7/x) * cos(7 ln x)
Therefore, the derivative of f(x) = sin(7 ln x) is f'(x) = (7/x) * cos(7 ln x).
To differentiate the function f(x) = sin(7 ln x), we can use the chain rule. The chain rule states that if we have a function, let's call it u, inside another function, let's call it v, then the derivative of v(u(x)) with respect to x is equal to the derivative of u with respect to v multiplied by the derivative of v with respect to x.
In this case, we have u(x) = 7 ln x inside the function v(x) = sin(x). To differentiate f(x), we need to find the derivatives of u(x) and v(x), and then apply the chain rule.
Step 1: Find the derivative of u(x) = 7 ln x with respect to x.
To find the derivative of ln x, we can use the formula:
d/dx(ln x) = 1/x
Therefore, the derivative of u(x) = 7 ln x with respect to x is:
du/dx = 7/x
Step 2: Find the derivative of v(x) = sin(x) with respect to x.
The derivative of sin(x) is:
dv/dx = cos(x)
Step 3: Apply the chain rule.
Using the chain rule, the derivative of f(x) = sin(7 ln x) is given by:
df/dx = dv/du * du/dx
Substituting the derivatives we found earlier, we have:
df/dx = cos(u(x)) * 7/x
Finally, we can simplify the expression to:
df/dx = (7/x) * cos(7 ln x)
So, the derivative of f(x) = sin(7 ln x) is (7/x) * cos(7 ln x).