Differentiate. \( f(x)=(\cos 6 x)^{7} \) \( f^{\prime}(x)=42(\cos 6 x)^{6}(\sin 6 x) \) \( f^{\prime}(x)=-42(\cos 6 x)^{6}(\sin 6 x) \) \( f^{\prime}(x)=-7(\cos 6 x)^{6}(\sin 6 x) \) \( f^{\prime}(x)=42(\cos 6 x)^{6} \)

To differentiate \( f(x) = (\cos 6x)^{7} \), you would apply the chain rule. First, recognize the outer function \( u^7 \) and the inner function \( u = \cos 6x \). The derivative of \( u^n \) is \( n u^{n-1} \cdot u' \). The inner function’s derivative \( u' \) is \( -6 \sin 6x \). Combining these gives: \[ f'(x) = 7(\cos 6x)^{6} \cdot (-6\sin 6x) = -42(\cos 6x)^{6}(\sin 6x) \] So the correct derivative is \( f^{\prime}(x) = -42(\cos 6 x)^{6}(\sin 6 x) \). If you’re tackling calculus problems, remember that clear organization of your workings can prevent you from getting lost in complex derivatives! Write down each step clearly and check your functions carefully. You can also utilize software tools like graphing calculators to visualize functions and their derivatives, helping you spot errors or confirm your answers. To dive deeper into this fascinating world, consider exploring resources about chain rule applications and trigonometric derivatives. Books like “Calculus Made Easy” by Silvanus P. Thompson or online platforms like Khan Academy can provide great insight. They break down challenging concepts into manageable bites, making the learning process both fun and engaging!