Question

hw12 derivatives of trigonometric functions (target c3; §3.5)
score: 1/7 answered: 1/7
question 2
if f(x)=5 sin x + 9 cos x, then
f(x) =
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Explanation:

Step1: Recall derivative rules

The derivative of $\sin x$ is $\cos x$ and the derivative of $\cos x$ is $-\sin x$. Also, for a function of the form $af(x)+bg(x)$ where $a$ and $b$ are constants, the derivative is $af'(x)+bg'(x)$ by the sum - rule of derivatives.

Step2: Differentiate each term

For the function $f(x)=5\sin x + 9\cos x$, the derivative of $5\sin x$ is $5\cos x$ (since the derivative of $\sin x$ is $\cos x$ and we multiply by the constant 5), and the derivative of $9\cos x$ is $9\times(-\sin x)=- 9\sin x$ (since the derivative of $\cos x$ is $-\sin x$ and we multiply by the constant 9).

Step3: Combine the derivatives

Using the sum - rule, $f'(x)=(5\sin x + 9\cos x)'=5\cos x-9\sin x$.

Answer:

$5\cos x - 9\sin x$