Question

the derivative of f(x) = 5 cos(2 - 3x) is f(x) = c sin(2 - 3x). find the value of c. give an exact answer as an integer. type here

Explanation:

Step1: Differentiate \(5\cos(2 - 3x)\)

Use the chain rule \((u(v(x)))^\prime=u^\prime(v(x))\cdot v^\prime(x)\). Let \(u = 5\cos t\) and \(t=2 - 3x\). Then \(u^\prime=- 5\sin t\) and \(t^\prime=-3\). So \((5\cos(2 - 3x))^\prime=-5\sin(2 - 3x)\cdot(-3)=15\sin(2 - 3x)\)

Step2: Compare with \(c\sin(2 - 3x)\)

Since \(f^\prime(x)=15\sin(2 - 3x)\) and \(f^\prime(x)=c\sin(2 - 3x)\), then \(c = 15\)

Answer:

\(15\)