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Find derivative b) f(x)=(x^(4)+3x^(2)-1)(x^(3)-5)

Find derivative\newlineb) f(x)=(x4+3x2−1)(x3−5) f(x)=\left(x^{4}+3 x^{2}-1\right)\left(x^{3}-5\right)

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Q. Find derivative\newlineb) f(x)=(x4+3x2−1)(x3−5) f(x)=\left(x^{4}+3 x^{2}-1\right)\left(x^{3}-5\right)
  1. Identify function: Identify the function to differentiate. The function is f(x)=(x4+3x2−1)(x3−5)f(x)=(x^{4}+3x^{2}-1)(x^{3}-5).
  2. Apply product rule: Apply the product rule for differentiation. The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. Let u(x)=x4+3x2−1u(x) = x^{4}+3x^{2}-1 and v(x)=x3−5v(x) = x^{3}-5. Then f′(x)=u′(x)v(x)+u(x)v′(x)f'(x) = u'(x)v(x) + u(x)v'(x).
  3. Find u′(x)u'(x): Find the derivative of u(x)=x4+3x2−1u(x) = x^{4}+3x^{2}-1. The derivative u′(x)u'(x) is found by differentiating each term separately: u′(x)=4x3+6xu'(x) = 4x^{3}+6x.
  4. Find v′(x)v'(x): Find the derivative of v(x)=x3−5v(x) = x^{3}-5. The derivative v′(x)v'(x) is found by differentiating each term separately: v′(x)=3x2v'(x) = 3x^{2}.
  5. Substitute derivatives: Substitute the derivatives u′(x)u'(x) and v′(x)v'(x) into the product rule formula. f′(x)=(4x3+6x)(x3−5)+(x4+3x2−1)(3x2)f'(x) = (4x^{3}+6x)(x^{3}-5) + (x^{4}+3x^{2}-1)(3x^{2}).
  6. Expand expressions: Expand the expressions in the formula. f′(x)=4x3∗x3−20x3+6x∗x3−30x+3x2∗x4+9x2∗x2−3x2.f'(x) = 4x^{3}*x^{3} - 20x^{3} + 6x*x^{3} - 30x + 3x^{2}*x^{4} + 9x^{2}*x^{2} - 3x^{2}.
  7. Simplify terms: Simplify the terms in the expression. f′(x)=4x6−20x3+6x4−30x+3x6+9x4−3x2f'(x) = 4x^{6} - 20x^{3} + 6x^{4} - 30x + 3x^{6} + 9x^{4} - 3x^{2}.
  8. Combine like terms: Combine like terms. f′(x)=(4x6+3x6)+(6x4+9x4)−20x3−3x2−30xf'(x) = (4x^{6} + 3x^{6}) + (6x^{4} + 9x^{4}) - 20x^{3} - 3x^{2} - 30x.
  9. Final simplification: Final simplification. f′(x)=7x6+15x4−20x3−3x2−30xf'(x) = 7x^{6} + 15x^{4} - 20x^{3} - 3x^{2} - 30x.

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