Home > Calculus > Chain Rule

How do you differentiate #f(x)=4(x^2 + x - 1)^10 # using the chain rule?

Answer 1

Isabella Ackerman

#f'(x)=40(x^2+x-1)^9(2x+1)#

We used that #(x^n)'=nx^{n-1}# and that #(x^2+x-1)'=2x+1#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Christopher Allers

To differentiate ( f(x) = 4(x^2 + x - 1)^{10} ) using the chain rule:

Identify the outer function ( u ) and the inner function ( v ). ( u = 4x^10 ) and ( v = x^2 + x - 1 ).
Compute the derivatives of ( u ) and ( v ): ( \frac{du}{dx} = 40(x^2 + x - 1)^9 ) and ( \frac{dv}{dx} = 2x + 1 ).
Apply the chain rule: ( \frac{df}{dx} = \frac{du}{dv} \times \frac{dv}{dx} ).
Substitute the derivatives and the functions: ( \frac{df}{dx} = 40(x^2 + x - 1)^9 \times (2x + 1) ).

By signing up, you agree to our Terms of Service and Privacy Policy

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.