1. **State the problem:** We need to find the derivative of the function $$f(x)=e^{\sin^3(7x)}$$. 2. **Rewrite the function:** The function can be written as $$f(x)=e^{(\sin(7x))^3}$$ to clearly indicate the cubic power on the sine function. 3. **Apply the chain rule:** The derivative of $$e^{u}$$ with respect to $$x$$ is $$e^{u} \cdot \frac{du}{dx}$$. Here, $$u=(\sin(7x))^3$$. 4. **Differentiate the inner function $$u$$:** - $$\frac{du}{dx} = 3(\sin(7x))^2 \cdot \frac{d}{dx}[\sin(7x)]$$ by the power rule. 5. **Differentiate $$\sin(7x)$$:** - $$\frac{d}{dx}[\sin(7x)] = \cos(7x) \cdot \frac{d}{dx}[7x] = \cos(7x) \cdot 7 = 7\cos(7x)$$. 6. **Combine the derivatives:** - $$\frac{du}{dx} = 3(\sin(7x))^2 \cdot 7\cos(7x) = 21 (\sin(7x))^2 \cos(7x)$$. 7. **Write the final derivative:** - $$f'(x) = e^{(\sin(7x))^3} \cdot 21 (\sin(7x))^2 \cos(7x)$$. **Final answer:** $$f'(x) = 21 e^{(\sin(7x))^3} (\sin(7x))^2 \cos(7x)$$