How do you find the derivative of #f(x)=10cos^3(4x)#?

Question

Evelyn Agard · Accepted Answer

#f'(x) = - 120 cos^2 (4x) * sin (4x)# 10 is constant take x, multiply by 4, take the cos, power 3 you derivate down upwards: #f'(x) = 10 * 3 cos^2 (4x) * (-sin (4x)) * 4#

Adam Adkins · Answer

undefined To find the derivative of $ f(x) = 10\cos^3(4x) $, you can use the chain rule and the power rule for differentiation. Here's the process:

1. Apply the chain rule: 
$$ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) $$

2. Let $ u = 4x $ and $ f(u) = 10\cos^3(u) $. Then $ f'(u) = -30\cos^2(u)\sin(u) $ by applying the power rule and the derivative of cosine.

3. Now differentiate $ u = 4x $ to find $ \frac{du}{dx} $:
$$ \frac{du}{dx} = 4 $$

4. Substitute $ f'(u) $ and $ \frac{du}{dx} $ into the chain rule formula:
$$ f'(x) = f'(u) \cdot \frac{du}{dx} = -30\cos^2(4x)\sin(4x) \cdot 4 $$

5. Simplify:
$$ f'(x) = -120\cos^2(4x)\sin(4x) $$

So, the derivative of $ f(x) = 10\cos^3(4x) $ is $ f'(x) = -120\cos^2(4x)\sin(4x) $.