Differentiate the function.
f(x) = sin(5 ln x)
the answer is:
(5/x)*(cos(5ln(x)))
In this problem:
1. Take the derivative of the cosine
2. Then take the derivative of whatever is inside the cosine. In this case (5ln(x))
3. Do not change whatever is inside the cosine.
i mean sin wherever it says cosine. My bad
To differentiate the function f(x) = sin(5 ln x), we can use the chain rule.
Step 1: Let's start by finding the derivative of the inner function, 5 ln x.
- The derivative of ln x with respect to x is 1/x.
- So, the derivative of 5 ln x with respect to x is 5/x.
Step 2: Now, let's differentiate the outer function, sin(u), where u = 5 ln x.
- The derivative of sin u with respect to u is cos u.
- So, the derivative of sin(5 ln x) with respect to x is cos(5 ln x) * (derivative of 5 ln x with respect to x).
Step 3: Substitute the derivative of 5 ln x from Step 1 into Step 2:
- The derivative of sin(5 ln x) with respect to x is cos(5 ln x) * (5/x).
Therefore, the derivative of f(x) = sin(5 ln x) is:
f'(x) = cos(5 ln x) * (5/x).
To differentiate the given function f(x) = sin(5 ln x), we can use the chain rule.
The chain rule states that if we have a composite function f(g(x)), where f is an outer function and g is an inner function, then the derivative is given by the product of the derivative of the outer function with respect to the inner function times the derivative of the inner function with respect to x.
Here's how we can apply the chain rule to differentiate f(x) = sin(5 ln x):
1. Start by differentiating the inner function g(x) = 5 ln x with respect to x:
g'(x) = 5/x
2. Next, differentiate the outer function f(u) = sin(u) with respect to u:
f'(u) = cos(u)
3. Apply the chain rule by multiplying the derivatives found in steps 1 and 2:
f'(x) = f'(u) * g'(x)
= cos(u) * (5/x)
4. Substitute u with the inner function g(x):
f'(x) = cos(5 ln x) * (5/x)
So, the derivative of f(x) = sin(5 ln x) is f'(x) = cos(5 ln x) * (5/x).