find the derivative of sin((3pi x)/25)cos((pi x)/25)+15

${\displaystyle \sin \left(\frac{3\pi x}{25}\right)\cos \left(\frac{\pi x}{25}\right)+15}$

By the Sum Rule, the derivative of ${\displaystyle \sin \left(\frac{3\pi x}{25}\right)\cos \left(\frac{\pi x}{25}\right)+15}$ with respect to ${\displaystyle x}$ is ${\displaystyle \frac{d}{dx}\left[\sin \left(\frac{3\pi x}{25}\right)\cos \left(\frac{\pi x}{25}\right)\right]+\frac{d}{dx}\left[15\right]}$.

${\displaystyle \frac{d}{dx}\left[\sin \left(\frac{3\pi x}{25}\right)\cos \left(\frac{\pi x}{25}\right)\right]+\frac{d}{dx}\left[15\right]}$

Evaluate ${\displaystyle \frac{d}{dx}\left[\sin \left(\frac{3\pi x}{25}\right)\cos \left(\frac{\pi x}{25}\right)\right]}$.

Differentiate using the Product Rule which states that ${\displaystyle \frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]}$ is ${\displaystyle f\left(x\right)\frac{d}{dx}\left[g\left(x\right)\right]+g\left(x\right)\frac{d}{dx}\left[f\left(x\right)\right]}$ where ${\displaystyle f\left(x\right)=\sin \left(\frac{3\pi x}{25}\right)}$ and ${\displaystyle g\left(x\right)=\cos \left(\frac{\pi x}{25}\right)}$.

${\displaystyle \sin \left(\frac{3\pi x}{25}\right)\frac{d}{dx}\left[\cos \left(\frac{\pi x}{25}\right)\right]+\cos \left(\frac{\pi x}{25}\right)\frac{d}{dx}\left[\sin \left(\frac{3\pi x}{25}\right)\right]+\frac{d}{dx}\left[15\right]}$

Differentiate using the chain rule, which states that ${\displaystyle \frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]}$ is ${\displaystyle f\prime \left(g\left(x\right)\right)g\prime \left(x\right)}$ where ${\displaystyle f\left(x\right)=\cos \left(x\right)}$ and ${\displaystyle g\left(x\right)=\frac{\pi x}{25}}$.

sin(3πx25)(−sin(πx25)ddx[πx25])+cos(πx25)ddx[sin(3πx25)]+ddx[15]${\displaystyle \sin \left(\frac{3\pi x}{25}\right)(-\sin \left(\frac{\pi x}{25}\right)\frac{d}{dx}\left[\frac{\pi x}{25}\right])+\cos \left(\frac{\pi x}{25}\right)\frac{d}{dx}\left[\sin \left(\frac{3\pi x}{25}\right)\right]+\frac{d}{dx}\left[15\right]}$

Since π25${\displaystyle \frac{\pi }{25}}$ is constant with respect to x${\displaystyle x}$, the derivative of πx25${\displaystyle \frac{\pi x}{25}}$ with respect to x${\displaystyle x}$ is π25ddx[x]${\displaystyle \frac{\pi }{25}\frac{d}{dx}\left[x\right]}$.

${\displaystyle \sin \left(\frac{3\pi x}{25}\right)(-\sin \left(\frac{\pi x}{25}\right)\left(\frac{\pi }{25}\frac{d}{dx}\left[x\right]\right))+\cos \left(\frac{\pi x}{25}\right)\frac{d}{dx}\left[\sin \left(\frac{3\pi x}{25}\right)\right]+\frac{d}{dx}\left[15\right]}$

Differentiate using the Power Rule which states that ${\displaystyle \frac{d}{dx}\left[x^n\right]}$ is ${\displaystyle n{x}^{n-1}}$ where ${\displaystyle n=1}$.

${\displaystyle \sin \left(\frac{3\pi x}{25}\right)(-\sin \left(\frac{\pi x}{25}\right)(\frac{\pi }{25}\cdot 1))+\cos \left(\frac{\pi x}{25}\right)\frac{d}{dx}\left[\sin \left(\frac{3\pi x}{25}\right)\right]+\frac{d}{dx}\left[15\right]}$

Differentiate using the chain rule, which states that ${\displaystyle \frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]}$ is ${\displaystyle f\prime \left(g\left(x\right)\right)g\prime \left(x\right)}$ where ${\displaystyle f\left(x\right)=\sin \left(x\right)}$ and ${\displaystyle g\left(x\right)=\frac{3\pi x}{25}}$.

${\displaystyle \sin \left(\frac{3\pi x}{25}\right)(-\sin \left(\frac{\pi x}{25}\right)(\frac{\pi }{25}\cdot 1))+\cos \left(\frac{\pi x}{25}\right)\left(\cos \left(\frac{3\pi x}{25}\right)\frac{d}{dx}\left[\frac{3\pi x}{25}\right]\right)+\frac{d}{dx}\left[15\right]}$

Since ${\displaystyle \frac{3\pi }{25}}$ is constant with respect to ${\displaystyle x}$, the derivative of ${\displaystyle \frac{3\pi x}{25}}$ with respect to ${\displaystyle x}$ is ${\displaystyle \frac{3\pi }{25}\frac{d}{dx}\left[x\right]}$.

${\displaystyle \sin \left(\frac{3\pi x}{25}\right)(-\sin \left(\frac{\pi x}{25}\right)(\frac{\pi }{25}\cdot 1))+\cos \left(\frac{\pi x}{25}\right)\left(\cos \left(\frac{3\pi x}{25}\right)\left(\frac{3\pi }{25}\frac{d}{dx}\left[x\right]\right)\right)+\frac{d}{dx}\left[15\right]}$

Differentiate using the Power Rule which states that ${\displaystyle \frac{d}{dx}\left[x^n\right]}$ is ${\displaystyle n{x}^{n-1}}$ where ${\displaystyle n=1}$.

${\displaystyle \sin \left(\frac{3\pi x}{25}\right)(-\sin \left(\frac{\pi x}{25}\right)(\frac{\pi }{25}\cdot 1))+\cos \left(\frac{\pi x}{25}\right)\left(\cos \left(\frac{3\pi x}{25}\right)(\frac{3\pi }{25}\cdot 1)\right)+\frac{d}{dx}\left[15\right]}$

Multiply ${\displaystyle \frac{\pi }{25}}$ by ${\displaystyle 1}$.

${\displaystyle \sin \left(\frac{3\pi x}{25}\right)(-\sin \left(\frac{\pi x}{25}\right)\frac{\pi }{25})+\cos \left(\frac{\pi x}{25}\right)\left(\cos \left(\frac{3\pi x}{25}\right)(\frac{3\pi }{25}\cdot 1)\right)+\frac{d}{dx}\left[15\right]}$

Combine ${\displaystyle \frac{\pi }{25}}$ and ${\displaystyle \sin \left(\frac{\pi x}{25}\right)}$.

${\displaystyle \sin \left(\frac{3\pi x}{25}\right)(-\frac{\pi \sin \left(\frac{\pi x}{25}\right)}{25})+\cos \left(\frac{\pi x}{25}\right)\left(\cos \left(\frac{3\pi x}{25}\right)(\frac{3\pi }{25}\cdot 1)\right)+\frac{d}{dx}\left[15\right]}$

Combine ${\displaystyle \sin \left(\frac{3\pi x}{25}\right)}$ and ${\displaystyle \frac{\pi \sin \left(\frac{\pi x}{25}\right)}{25}}$.

${\displaystyle -\frac{\sin \left(\frac{3\pi x}{25}\right)\left(\pi \sin \left(\frac{\pi x}{25}\right)\right)}{25}+\cos \left(\frac{\pi x}{25}\right)\left(\cos \left(\frac{3\pi x}{25}\right)(\frac{3\pi }{25}\cdot 1)\right)+\frac{d}{dx}\left[15\right]}$

Multiply ${\displaystyle \frac{3\pi }{25}}$ by ${\displaystyle 1}$.

${\displaystyle -\frac{\sin \left(\frac{3\pi x}{25}\right)\pi \sin \left(\frac{\pi x}{25}\right)}{25}+\cos \left(\frac{\pi x}{25}\right)\left(\cos \left(\frac{3\pi x}{25}\right)\frac{3\pi }{25}\right)+\frac{d}{dx}\left[15\right]}$

Combine ${\displaystyle \cos \left(\frac{3\pi x}{25}\right)}$ and ${\displaystyle \frac{3\pi }{25}}$.

${\displaystyle -\frac{\sin \left(\frac{3\pi x}{25}\right)\pi \sin \left(\frac{\pi x}{25}\right)}{25}+\cos \left(\frac{\pi x}{25}\right)\frac{\cos \left(\frac{3\pi x}{25}\right)\left(3\pi \right)}{25}+\frac{d}{dx}\left[15\right]}$

Move ${\displaystyle 3}$ to the left of ${\displaystyle \cos \left(\frac{3\pi x}{25}\right)}$.

${\displaystyle -\frac{\sin \left(\frac{3\pi x}{25}\right)\pi \sin \left(\frac{\pi x}{25}\right)}{25}+\cos \left(\frac{\pi x}{25}\right)\frac{3\cdot \cos \left(\frac{3\pi x}{25}\right)\pi }{25}+\frac{d}{dx}\left[15\right]}$

Combine ${\displaystyle \cos \left(\frac{\pi x}{25}\right)}$ and ${\displaystyle \frac{3\cos \left(\frac{3\pi x}{25}\right)\pi }{25}}$.

${\displaystyle -\frac{\sin \left(\frac{3\pi x}{25}\right)\pi \sin \left(\frac{\pi x}{25}\right)}{25}+\frac{\cos \left(\frac{\pi x}{25}\right)\left(3\cos \left(\frac{3\pi x}{25}\right)\pi \right)}{25}+\frac{d}{dx}\left[15\right]}$

Move ${\displaystyle 3}$ to the left of ${\displaystyle \cos \left(\frac{\pi x}{25}\right)}$.

${\displaystyle -\frac{\sin \left(\frac{3\pi x}{25}\right)\pi \sin \left(\frac{\pi x}{25}\right)}{25}+\frac{3\cos \left(\frac{\pi x}{25}\right)\cos \left(\frac{3\pi x}{25}\right)\pi }{25}+\frac{d}{dx}\left[15\right]}$

Since ${\displaystyle 15}$ is constant with respect to ${\displaystyle x}$, the derivative of ${\displaystyle 15}$ with respect to ${\displaystyle x}$ is ${\displaystyle 0}$.

${\displaystyle -\frac{\sin \left(\frac{3\pi x}{25}\right)\pi \sin \left(\frac{\pi x}{25}\right)}{25}+\frac{3\cos \left(\frac{\pi x}{25}\right)\cos \left(\frac{3\pi x}{25}\right)\pi }{25}+0}$

Simplify.

${\displaystyle -\frac{\sin \left(\frac{3\pi x}{25}\right)\pi \sin \left(\frac{\pi x}{25}\right)}{25}+\frac{3\cos \left(\frac{\pi x}{25}\right)\cos \left(\frac{3\pi x}{25}\right)\pi }{25}}$ and ${\displaystyle 0}$.

${\displaystyle -\frac{\sin \left(\frac{3\pi x}{25}\right)\pi \sin \left(\frac{\pi x}{25}\right)}{25}+\frac{3\cos \left(\frac{\pi x}{25}\right)\cos \left(\frac{3\pi x}{25}\right)\pi }{25}}$

Reorder terms.

${\displaystyle \frac{3\cos \left(\frac{\pi x}{25}\right)\cos \left(\frac{3\pi x}{25}\right)\pi }{25}-\frac{\sin \left(\frac{3\pi x}{25}\right)\pi \sin \left(\frac{\pi x}{25}\right)}{25}}$

Reorder factors in ${\displaystyle \frac{3\cos \left(\frac{\pi x}{25}\right)\cos \left(\frac{3\pi x}{25}\right)\pi }{25}-\frac{\sin \left(\frac{3\pi x}{25}\right)\pi \sin \left(\frac{\pi x}{25}\right)}{25}}$.

$\frac{3\pi \cos \left(\frac{\pi x}{25}\right)\cos \left(\frac{3\pi x}{25}\right)}{25}-\frac{\pi \sin \left(\frac{3\pi x}{25}\right)\sin \left(\frac{\pi x}{25}\right)}{25}$ - Answer