difertiate g(x)=2cos(-5x)
explain steps
To differentiate the function g(x) = 2cos(-5x), we can use the chain rule.
The chain rule states that if we have a composite function, f(g(x)), then the derivative of that function can be found by taking the derivative of the outer function, f'(g(x)), and multiplying it by the derivative of the inner function, g'(x).
In this case, our outer function is f(x) = 2cos(x) and our inner function is g(x) = -5x.
The derivative of the outer function f(x) = 2cos(x) is f'(x) = -2sin(x), which we can obtain by using the derivative of cosine.
The derivative of the inner function g(x) = -5x is g'(x) = -5, as the derivative of -5x is simply -5.
Now we can apply the chain rule by multiplying the derivatives of the outer and inner functions:
g'(x) = f'(g(x)) * g'(x) = -2sin(-5x) * -5
Simplifying further,
g'(x) = 10sin(-5x)
Thus, the derivative of g(x) = 2cos(-5x) is g'(x) = 10sin(-5x).
To differentiate the function g(x) = 2cos(-5x), we will use the chain rule. The chain rule states that for a composition of functions, d/dx (f(g(x))) = f'(g(x)) * g'(x).
Step 1: Identify the inner function
In this case, the inner function is -5x.
Step 2: Calculate the derivative of the inner function
Since the derivative of -5x with respect to x is -5, g'(x) = -5.
Step 3: Calculate the derivative of the outer function
The outer function is cos(x). The derivative of cos(x) with respect to x is -sin(x). However, since we have a constant coefficient of 2, we multiply by 2 to get g'(x) = -2sin(x).
Step 4: Combine the derivatives using the chain rule
Using the chain rule formula, we have d/dx (2cos(-5x)) = -2sin(-5x) * -5.
Step 5: Simplify the expression
Since sin(-x) = -sin(x), we have d/dx (2cos(-5x)) = -2sin(-5x) * -5 = 10sin(5x).
So, the derivative of g(x) = 2cos(-5x) is 10sin(5x).
To differentiate the function g(x) = 2cos(-5x), you can use the chain rule of differentiation. The chain rule is used when you have a composition of functions, where one function is inside another.
Here are the steps to differentiate g(x) = 2cos(-5x):
Step 1: Identify the inner function and its derivative.
The inner function in this case is -5x. The derivative of -5x with respect to x is -5.
Step 2: Differentiate the outer function.
The outer function is cos(x). The derivative of cos(x) with respect to x is -sin(x).
Step 3: Apply the chain rule.
To apply the chain rule, multiply the derivative of the outer function (-sin(x)) by the derivative of the inner function (-5) and multiply by the original coefficient (2).
Therefore, the derivative of g(x) = 2cos(-5x) is:
g'(x) = (2) * (-sin(-5x)) * (-5)
Now simplify the expression:
g'(x) = 10sin(-5x)
So, the derivative of g(x) = 2cos(-5x) is g'(x) = 10sin(-5x).