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If #f(x)= 2 x^2 - 3 x # and #g(x) = 2e^-x -4 #, how do you differentiate #f(g(x)) # using the chain rule?

Answer 1

Ezekiel Alexander

See the answer below:

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Answer 2

Ezekiel Alexander

To differentiate ( f(g(x)) ) using the chain rule, follow these steps:

Find the derivative of ( f(x) ) with respect to ( x ), which is ( f'(x) ).
Find the derivative of ( g(x) ) with respect to ( x ), which is ( g'(x) ).
Substitute ( g(x) ) into ( f(x) ) to get ( f(g(x)) ).
Apply the chain rule: ( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) ).

Given ( f(x) = 2x^2 - 3x ) and ( g(x) = 2e^{-x} - 4 ):

Find ( f'(x) ): ( f'(x) = \frac{d}{dx}[2x^2 - 3x] = 4x - 3 ).
Find ( g'(x) ): ( g'(x) = \frac{d}{dx}[2e^{-x} - 4] = -2e^{-x} ).
Substitute ( g(x) ) into ( f(x) ) to get ( f(g(x)) ): ( f(g(x)) = 2(2e^{-x} - 4)^2 - 3(2e^{-x} - 4) ).
Apply the chain rule: ( \frac{d}{dx}[f(g(x))] = (4(2e^{-x} - 4) - 3)(-2e^{-x}) ).

Simplify the expression to get the final result.

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Answer from HIX Tutor

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.