What is the derivative of #(x-1)(x^2+2)^3#?

Answer 1

#(x^2+2)^2(7x^2-6x+2)#

#"differentiate using the "color(blue)"product rule"#
#"given "y=g(x).h(x)" then"#
#dy/dx=g(x)h'(x)+h(x)g'(x)larr" product rule"#
#g(x)=x-1rArrg'(x)=1#
#h(x)=(x^2+2)^3rArrh'(x)=3(x^2+2)^2.d/dx(x^2+2)#
#color(white)(xxxxxxxxxxxxxxxxxx)=6x(x^2+2)^2#
#rArrdy/dx=6x(x-1)(x^2+2)^2+(x^2+2)^3#
#color(white)(rArrdy/dx)=(x^2+2)^2(6x^2-6x+x^2+2)#
#color(white)(rArrdy/dx)=(x^2+2)^2(7x^2-6x+2)#
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Answer 2

To find the derivative of ( (x-1)(x^2+2)^3 ):

  1. Identify the functions being multiplied: ( f(x) = x - 1 ) and ( g(x) = (x^2 + 2)^3 ).
  2. Apply the product rule: ( (fg)'(x) = f'(x)g(x) + f(x)g'(x) ).
  3. Find the derivatives of ( f(x) ) and ( g(x) ):
    • ( f'(x) = 1 )
    • ( g'(x) = 3(x^2 + 2)^2 \cdot (2x) ).
  4. Substitute the derivatives and functions into the product rule formula.
  5. Simplify the expression.
  6. Combine like terms if necessary 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.

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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