Question

Find the derivative of y=\sin 2x +3\cos 5x

Answer 1

Answer on Question #82957 – Math – Differential Equations

Question

Find the derivative of $y = \sin 2x + 3\cos 5x$.

Solution

Use the Sum Rule, that is:

$$(f(x) + g(x))' = f'(x) + g'(x),$$

and the Chain Rule:

$$\text{if } f(x) = g[h(x)], \text{ then } f'(x) = g'[h(x)] \cdot h'(x).$$

Use derivatives of trigonometric functions $\sin x$ and $\cos x$:

$$(\sin x)' = \cos x, \text{ and } (\cos x)' = -\sin x.$$

Find the derivative of $y'$:

$$y' = (\sin 2x)' \cdot (2x)' + (3\cos 5x)' \cdot (5x)' = \cos 2x \cdot 2 - 3\sin 5x \cdot 5 = 2\cos 2x - 15\sin 5x.$$

Answer: $y' = 2\cos 2x - 15\sin 5x$.

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