What is the derivative of 5cos(x)?
Understand the Problem
The question is asking for the derivative of the function 5cos(x) with respect to x. To solve it, we will apply the derivative rules, specifically the constant multiple rule and the derivative of cosine.
Answer
The derivative is $f'(x) = -5\sin(x)$.
Answer for screen readers
The derivative of the function $5\cos(x)$ with respect to $x$ is $f'(x) = -5\sin(x)$.
Steps to Solve
- Identify the function and rules to use
We are given the function $f(x) = 5\cos(x)$. We will use the derivative rules, specifically the constant multiple rule and the derivative of the cosine function.
- Apply the constant multiple rule
According to the constant multiple rule, the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. Thus, we have: $$ f'(x) = 5 \cdot \frac{d}{dx}[\cos(x)] $$
- Find the derivative of cosine
The derivative of $\cos(x)$ is $-\sin(x)$. Therefore, we can substitute this into our previous equation: $$ f'(x) = 5 \cdot (-\sin(x)) $$
- Simplify the expression
Now we simplify the expression: $$ f'(x) = -5\sin(x) $$
The derivative of the function $5\cos(x)$ with respect to $x$ is $f'(x) = -5\sin(x)$.
More Information
The process of finding the derivative of a cosine function is a fundamental aspect of calculus. The negative sign that appears when differentiating the cosine function is a critical point often emphasized in derivative rules. Understanding these derivatives is crucial for studying wave mechanics and oscillatory behavior in physics.
Tips
- Forgetting to apply the negative sign when differentiating cosine. Always remember that the derivative of $\cos(x)$ is $-\sin(x)$.
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