How do you find the derivative of #f(x)=10cos^3(4x)#?
10 is constant take x, multiply by 4, take the cos, power 3
you derivate down upwards:
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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:

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

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.

Now differentiate ( u = 4x ) to find ( \frac{du}{dx} ): [ \frac{du}{dx} = 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 ]

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) ).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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