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Derivative of $$$\sin{\left(5 \theta \right)}$$$
Related calculators: Logarithmic Differentiation Calculator, Implicit Differentiation Calculator with Steps
Your Input
Find $$$\frac{d}{d\theta} \left(\sin{\left(5 \theta \right)}\right)$$$.
Solution
The function $$$\sin{\left(5 \theta \right)}$$$ is the composition $$$f{\left(g{\left(\theta \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ and $$$g{\left(\theta \right)} = 5 \theta$$$.
Apply the chain rule $$$\frac{d}{d\theta} \left(f{\left(g{\left(\theta \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{d\theta} \left(g{\left(\theta \right)}\right)$$$:
$${\color{red}\left(\frac{d}{d\theta} \left(\sin{\left(5 \theta \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{d\theta} \left(5 \theta\right)\right)}$$The derivative of the sine is $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{d\theta} \left(5 \theta\right) = {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{d\theta} \left(5 \theta\right)$$Return to the old variable:
$$\cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{d\theta} \left(5 \theta\right) = \cos{\left({\color{red}\left(5 \theta\right)} \right)} \frac{d}{d\theta} \left(5 \theta\right)$$Apply the constant multiple rule $$$\frac{d}{d\theta} \left(c f{\left(\theta \right)}\right) = c \frac{d}{d\theta} \left(f{\left(\theta \right)}\right)$$$ with $$$c = 5$$$ and $$$f{\left(\theta \right)} = \theta$$$:
$$\cos{\left(5 \theta \right)} {\color{red}\left(\frac{d}{d\theta} \left(5 \theta\right)\right)} = \cos{\left(5 \theta \right)} {\color{red}\left(5 \frac{d}{d\theta} \left(\theta\right)\right)}$$Apply the power rule $$$\frac{d}{d\theta} \left(\theta^{n}\right) = n \theta^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{d\theta} \left(\theta\right) = 1$$$:
$$5 \cos{\left(5 \theta \right)} {\color{red}\left(\frac{d}{d\theta} \left(\theta\right)\right)} = 5 \cos{\left(5 \theta \right)} {\color{red}\left(1\right)}$$Thus, $$$\frac{d}{d\theta} \left(\sin{\left(5 \theta \right)}\right) = 5 \cos{\left(5 \theta \right)}$$$.
Answer
$$$\frac{d}{d\theta} \left(\sin{\left(5 \theta \right)}\right) = 5 \cos{\left(5 \theta \right)}$$$A