# How do we differentiate #f(x)=sin^4(4x^2-6x+1)# using chain rule?

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To differentiate ( f(x) = \sin^4(4x^2 - 6x + 1) ) using the chain rule:

- Identify the outer function and the inner function.
- Apply the chain rule, which states that if ( y = u^n ), where ( u ) is a function of ( x ), then ( \frac{dy}{dx} = n \cdot u^{(n-1)} \cdot \frac{du}{dx} ).
- Differentiate the outer function with respect to the inner function.
- Differentiate the inner function with respect to ( x ).
- Multiply the results of steps 3 and 4.

Applying these steps:

- Outer function: ( u = \sin^4(\cdot) )
- Inner function: ( v = 4x^2 - 6x + 1 )

Differentiating the outer function: ( \frac{du}{dv} = 4 \sin^3(v) \cdot \cos(v) )

Differentiating the inner function: ( \frac{dv}{dx} = 8x - 6 )

Multiplying the results: ( \frac{df}{dx} = 4 \sin^3(4x^2 - 6x + 1) \cdot \cos(4x^2 - 6x + 1) \cdot (8x - 6) )

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When evaluating a one-sided 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 one-sided 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 one-sided 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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