opriate rule to differentiate \[ y=\left(7 x^{4}+4 c^{-x}\right) \sin (5 x) \] \[ \frac{d y}{d x}=\frac{1}{4} \]

To differentiate the given function \( y = (7x^4 + 4c^{-x}) \sin(5x) \), you’ll want to use the product rule, which states that if you have two functions multiplied together, you differentiate the first function and multiply by the second, then add the first function multiplied by the derivative of the second. The first part involves differentiating \( 7x^4 + 4c^{-x} \), leading to \( 28x^3 - 4c^{-x}\ln(c) \). The second part involves differentiating \( \sin(5x) \) to get \( 5\cos(5x) \). Combining these using the product rule results in: \[ \frac{dy}{dx} = (28x^3 - 4c^{-x}\ln(c))\sin(5x) + (7x^4 + 4c^{-x}) \cdot 5\cos(5x) \] Setting \( \frac{dy}{dx} = \frac{1}{4} \) can allow you to solve for specific values of \( x \). To interpret the differentiation process funnily, consider that differentiating is like giving your function a wake-up call – you reminded it of how quickly things are changing in the world of mathematics! Now it’s all energized, ready to strut its derivative dance moves!