Question

if $f(x)=6sin x + 5cos x$, then $f(x)=-6cos x - 5sin x$

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 sum of functions $u(x)+v(x)$, the derivative is $u'(x)+v'(x)$ (sum - rule of derivatives).

Step2: Differentiate each term

For $u(x) = 6\sin x$, by the constant - multiple rule ($(cf(x))'=cf'(x)$ where $c = 6$ and $f(x)=\sin x$), $u'(x)=6\cos x$. For $v(x)=5\cos x$, by the constant - multiple rule ($c = 5$ and $f(x)=\cos x$), $v'(x)=- 5\sin x$.

Step3: Find the derivative of $f(x)$

Using the sum - rule $f'(x)=u'(x)+v'(x)$, so $f'(x)=6\cos x-5\sin x$.

Answer:

$6\cos x - 5\sin x$