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How do you find the derivative of #(x^4 - 1)^10 (2x^4 + 3)^7#?

Answer 1

Cameron Alexander

#56x^3(x^4-1)^10(2x^4+3)^6+40x^3(2x^4+3)^7(x^4-1)^9#

Use the product rule : #d/dx(f(x)*g(x))=f(x)*g'(x)+g(x)*f'(x)#

and inside this product rule there will be need for use of the power rule : #d/dx[u(x)]^n=n*[u(x)]^(n-1)*(du)/(dx)#.

Thus, the function's derivative with respect to x is as follows:

#(x^4-1)^10*7(2x^4+3)^6*(8x^3)+(2x^4+3)^7*10(x^4-1)^9*(4x^3)#

#=56x^3(x^4-1)^10(2x^4+3)^6+40x^3(2x^4+3)^7(x^4-1)^9#.

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Answer 2

William Abdalla

To find the derivative of ( (x^4 - 1)^{10} \times (2x^4 + 3)^7 ), you can use the chain rule and the product rule. The derivative is:

[ \frac{d}{dx}[(x^4 - 1)^{10} \times (2x^4 + 3)^7] = ] [ 10(x^4 - 1)^9 \times (2x^4 + 3)^7 \times \frac{d}{dx}(x^4 - 1) + 7(x^4 - 1)^{10} \times (2x^4 + 3)^6 \times \frac{d}{dx}(2x^4 + 3) ]

[ = 10(x^4 - 1)^9 \times (2x^4 + 3)^7 \times (4x^3) + 7(x^4 - 1)^{10} \times (2x^4 + 3)^6 \times (8x^3) ]

[ = 40x^3(x^4 - 1)^9 \times (2x^4 + 3)^7 + 56x^3(x^4 - 1)^{10} \times (2x^4 + 3)^6 ]

This is the derivative of the given expression.

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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.