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(5) If y = ln(5 + 2sin x), find dy/dx

June 5, 2026Mathematics

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Answer

\frac{2\cos x}{5 + 2\sin x}

Graph of y = ln(5 + 2sin(x))

4 steps

Identify the function and apply the chain rule.
Given $y = \ln(5 + 2\sin x)$ . Let $u = 5 + 2\sin x$ . Then $y = \ln u$ .
The chain rule states that $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ .
Differentiate $y$ with respect to $u$ .
$\frac{dy}{du} = \frac{d}{du}(\ln u) = \frac{1}{u}$
Differentiate $u$ with respect to $x$ .
$\frac{du}{dx} = \frac{d}{dx}(5 + 2\sin x)$
The derivative of a constant (5) is 0. The derivative of $2\sin x$ is $2\cos x$ .
$\frac{du}{dx} = 0 + 2\cos x = 2\cos x$
Substitute $u$ back and multiply the derivatives.
$\frac{dy}{dx} = \frac{1}{5 + 2\sin x} \cdot (2\cos x)$ $\frac{dy}{dx} = \frac{2\cos x}{5 + 2\sin x}$
The derivative is $\frac{2\cos x}{5 + 2\sin x}$ .

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