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Given the function f(x)=(1+3x^(-3)+6x^(-2))(-8x^(2)-6), find f^(')(x) in any form. Answer: f^(')(x)=

Given the function f(x)=(1+3x−3+6x−2)(−8x2−6) f(x)=\left(1+3 x^{-3}+6 x^{-2}\right)\left(-8 x^{2}-6\right) , find f′(x) f^{\prime}(x) in any form.\newlineAnswer: f′(x)= f^{\prime}(x)=

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Q. Given the function f(x)=(1+3x−3+6x−2)(−8x2−6) f(x)=\left(1+3 x^{-3}+6 x^{-2}\right)\left(-8 x^{2}-6\right) , find f′(x) f^{\prime}(x) in any form.\newlineAnswer: f′(x)= f^{\prime}(x)=
  1. Find Derivative of f(x)f(x): We need to find the derivative of the function f(x)f(x) which is a product of two functions: (1+3x−3+6x−2)(1+3x^{-3}+6x^{-2}) and (−8x2−6)(-8x^{2}-6). We will use the product rule which 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.
  2. Derivative of g(x)g(x): First, let's find the derivative of the first function, g(x)=1+3x−3+6x−2g(x) = 1+3x^{-3}+6x^{-2}. The derivative of a sum of functions is the sum of their derivatives. So we will find the derivatives of 11, 3x−33x^{-3}, and 6x−26x^{-2} separately.
  3. Derivative of h(x)h(x): The derivative of a constant is 00, so ddx(1)=0\frac{d}{dx}(1) = 0.
  4. Apply Product Rule: Using the power rule, (d)/(dx)(xn)=nx(n−1)(d)/(dx)(x^n) = nx^{(n-1)}, we find the derivative of 3x−33x^{-3} as follows:\newline(d)/(dx)(3x−3)=3×(−3)×x(−3−1)=−9x−4(d)/(dx)(3x^{-3}) = 3 \times (-3) \times x^{(-3-1)} = -9x^{-4}.
  5. Combine Derivatives: Similarly, we find the derivative of 6x−26x^{-2} as follows:\newlineddx(6x−2)=6⋅(−2)⋅x−2−1=−12x−3\frac{d}{dx}(6x^{-2}) = 6 \cdot (-2) \cdot x^{-2-1} = -12x^{-3}.
  6. Simplify Expression: Now we can combine the derivatives of the parts of the first function: g′(x)=ddx(1)+ddx(3x−3)+ddx(6x−2)=0−9x−4−12x−3g'(x) = \frac{d}{dx}(1) + \frac{d}{dx}(3x^{-3}) + \frac{d}{dx}(6x^{-2}) = 0 - 9x^{-4} - 12x^{-3}.
  7. Simplify Expression: Now we can combine the derivatives of the parts of the first function: g′(x)=ddx(1)+ddx(3x−3)+ddx(6x−2)=0−9x−4−12x−3g'(x) = \frac{d}{dx}(1) + \frac{d}{dx}(3x^{-3}) + \frac{d}{dx}(6x^{-2}) = 0 - 9x^{-4} - 12x^{-3}.Next, we find the derivative of the second function, h(x)=−8x2−6h(x) = -8x^{2}-6. The derivative of a constant times a function is the constant times the derivative of the function, and the derivative of a constant is 00.
  8. Simplify Expression: Now we can combine the derivatives of the parts of the first function: g′(x)=ddx(1)+ddx(3x−3)+ddx(6x−2)=0−9x−4−12x−3g'(x) = \frac{d}{dx}(1) + \frac{d}{dx}(3x^{-3}) + \frac{d}{dx}(6x^{-2}) = 0 - 9x^{-4} - 12x^{-3}. Next, we find the derivative of the second function, h(x)=−8x2−6h(x) = -8x^{2}-6. The derivative of a constant times a function is the constant times the derivative of the function, and the derivative of a constant is 00. Using the power rule again, we find the derivative of −8x2-8x^{2} as follows: ddx(−8x2)=−8×2×x2−1=−16x\frac{d}{dx}(-8x^{2}) = -8 \times 2 \times x^{2-1} = -16x.
  9. Simplify Expression: Now we can combine the derivatives of the parts of the first function: g′(x)=ddx(1)+ddx(3x−3)+ddx(6x−2)=0−9x−4−12x−3g'(x) = \frac{d}{dx}(1) + \frac{d}{dx}(3x^{-3}) + \frac{d}{dx}(6x^{-2}) = 0 - 9x^{-4} - 12x^{-3}.Next, we find the derivative of the second function, h(x)=−8x2−6h(x) = -8x^{2}-6. The derivative of a constant times a function is the constant times the derivative of the function, and the derivative of a constant is 00.Using the power rule again, we find the derivative of −8x2-8x^{2} as follows: ddx(−8x2)=−8×2×x2−1=−16x\frac{d}{dx}(-8x^{2}) = -8 \times 2 \times x^{2-1} = -16x.The derivative of −6-6 is 00, so ddx(−6)=0\frac{d}{dx}(-6) = 0.
  10. Simplify Expression: Now we can combine the derivatives of the parts of the first function: g′(x)=ddx(1)+ddx(3x−3)+ddx(6x−2)=0−9x−4−12x−3g'(x) = \frac{d}{dx}(1) + \frac{d}{dx}(3x^{-3}) + \frac{d}{dx}(6x^{-2}) = 0 - 9x^{-4} - 12x^{-3}.Next, we find the derivative of the second function, h(x)=−8x2−6h(x) = -8x^{2}-6. The derivative of a constant times a function is the constant times the derivative of the function, and the derivative of a constant is 00.Using the power rule again, we find the derivative of −8x2-8x^{2} as follows: ddx(−8x2)=−8×2×x2−1=−16x\frac{d}{dx}(-8x^{2}) = -8 \times 2 \times x^{2-1} = -16x.The derivative of −6-6 is 00, so ddx(−6)=0\frac{d}{dx}(-6) = 0.Now we can combine the derivatives of the parts of the second function: h′(x)=ddx(−8x2)+ddx(−6)=−16x+0=−16xh'(x) = \frac{d}{dx}(-8x^{2}) + \frac{d}{dx}(-6) = -16x + 0 = -16x.
  11. Simplify Expression: Now we can combine the derivatives of the parts of the first function: \newlineg′(x)=ddx(1)+ddx(3x−3)+ddx(6x−2)=0−9x−4−12x−3.g'(x) = \frac{d}{dx}(1) + \frac{d}{dx}(3x^{-3}) + \frac{d}{dx}(6x^{-2}) = 0 - 9x^{-4} - 12x^{-3}.Next, we find the derivative of the second function, h(x)=−8x2−6h(x) = -8x^{2}-6. The derivative of a constant times a function is the constant times the derivative of the function, and the derivative of a constant is 00.Using the power rule again, we find the derivative of −8x2-8x^{2} as follows:\newlineddx(−8x2)=−8×2×x2−1=−16x.\frac{d}{dx}(-8x^{2}) = -8 \times 2 \times x^{2-1} = -16x.The derivative of −6-6 is 00, so ddx(−6)=0\frac{d}{dx}(-6) = 0.Now we can combine the derivatives of the parts of the second function:\newlineh′(x)=ddx(−8x2)+ddx(−6)=−16x+0=−16x.h'(x) = \frac{d}{dx}(-8x^{2}) + \frac{d}{dx}(-6) = -16x + 0 = -16x.Using the product rule, we find the derivative of the product of g(x)g(x) and h(x)=−8x2−6h(x) = -8x^{2}-600:\newlineh(x)=−8x2−6h(x) = -8x^{2}-611
  12. Simplify Expression: Now we can combine the derivatives of the parts of the first function: g′(x)=ddx(1)+ddx(3x−3)+ddx(6x−2)=0−9x−4−12x−3.g'(x) = \frac{d}{dx}(1) + \frac{d}{dx}(3x^{-3}) + \frac{d}{dx}(6x^{-2}) = 0 - 9x^{-4} - 12x^{-3}. Next, we find the derivative of the second function, h(x)=−8x2−6h(x) = -8x^{2}-6. The derivative of a constant times a function is the constant times the derivative of the function, and the derivative of a constant is 00.Using the power rule again, we find the derivative of −8x2-8x^{2} as follows: ddx(−8x2)=−8×2×x2−1=−16x\frac{d}{dx}(-8x^{2}) = -8 \times 2 \times x^{2-1} = -16x. The derivative of −6-6 is 00, so ddx(−6)=0\frac{d}{dx}(-6) = 0. Now we can combine the derivatives of the parts of the second function: h′(x)=ddx(−8x2)+ddx(−6)=−16x+0=−16xh'(x) = \frac{d}{dx}(-8x^{2}) + \frac{d}{dx}(-6) = -16x + 0 = -16x. Using the product rule, we find the derivative of the product of g(x)g(x) and h(x)h(x): f′(x)=g′(x)h(x)+g(x)h′(x)f'(x) = g'(x)h(x) + g(x)h'(x). Substitute the derivatives and original functions into the product rule formula: h(x)=−8x2−6h(x) = -8x^{2}-600.
  13. Simplify Expression: Now we can combine the derivatives of the parts of the first function: \newlineg′(x)=ddx(1)+ddx(3x−3)+ddx(6x−2)=0−9x−4−12x−3.g'(x) = \frac{d}{dx}(1) + \frac{d}{dx}(3x^{-3}) + \frac{d}{dx}(6x^{-2}) = 0 - 9x^{-4} - 12x^{-3}. Next, we find the derivative of the second function, h(x)=−8x2−6h(x) = -8x^{2}-6. The derivative of a constant times a function is the constant times the derivative of the function, and the derivative of a constant is 00. Using the power rule again, we find the derivative of −8x2-8x^{2} as follows: \newlineddx(−8x2)=−8×2×x2−1=−16x.\frac{d}{dx}(-8x^{2}) = -8 \times 2 \times x^{2-1} = -16x. The derivative of −6-6 is 00, so ddx(−6)=0\frac{d}{dx}(-6) = 0. Now we can combine the derivatives of the parts of the second function: \newlineh′(x)=ddx(−8x2)+ddx(−6)=−16x+0=−16x.h'(x) = \frac{d}{dx}(-8x^{2}) + \frac{d}{dx}(-6) = -16x + 0 = -16x. Using the product rule, we find the derivative of the product of g(x)g(x) and h(x)=−8x2−6h(x) = -8x^{2}-600: \newlineh(x)=−8x2−6h(x) = -8x^{2}-611 Substitute the derivatives and original functions into the product rule formula: \newlineh(x)=−8x2−6h(x) = -8x^{2}-622 Now we need to simplify the expression. We will distribute the terms and combine like terms where possible. Let's start with the first part of the product rule: \newlineh(x)=−8x2−6h(x) = -8x^{2}-633
  14. Simplify Expression: Now we can combine the derivatives of the parts of the first function: \newlineg′(x)=ddx(1)+ddx(3x−3)+ddx(6x−2)=0−9x−4−12x−3.g'(x) = \frac{d}{dx}(1) + \frac{d}{dx}(3x^{-3}) + \frac{d}{dx}(6x^{-2}) = 0 - 9x^{-4} - 12x^{-3}.Next, we find the derivative of the second function, h(x)=−8x2−6h(x) = -8x^{2}-6. The derivative of a constant times a function is the constant times the derivative of the function, and the derivative of a constant is 00.Using the power rule again, we find the derivative of −8x2-8x^{2} as follows:\newlineddx(−8x2)=−8×2×x2−1=−16x.\frac{d}{dx}(-8x^{2}) = -8 \times 2 \times x^{2-1} = -16x.The derivative of −6-6 is 00, so ddx(−6)=0\frac{d}{dx}(-6) = 0.Now we can combine the derivatives of the parts of the second function:\newlineh′(x)=ddx(−8x2)+ddx(−6)=−16x+0=−16x.h'(x) = \frac{d}{dx}(-8x^{2}) + \frac{d}{dx}(-6) = -16x + 0 = -16x.Using the product rule, we find the derivative of the product of g(x)g(x) and h(x)=−8x2−6h(x) = -8x^{2}-600:\newlineh(x)=−8x2−6h(x) = -8x^{2}-611.Substitute the derivatives and original functions into the product rule formula:\newlineh(x)=−8x2−6h(x) = -8x^{2}-622.Now we need to simplify the expression. We will distribute the terms and combine like terms where possible. Let's start with the first part of the product rule:\newlineh(x)=−8x2−6h(x) = -8x^{2}-633.Now let's distribute the second part of the product rule:\newlineh(x)=−8x2−6h(x) = -8x^{2}-644.
  15. Simplify Expression: Now we can combine the derivatives of the parts of the first function: \newlineg′(x)=ddx(1)+ddx(3x−3)+ddx(6x−2)=0−9x−4−12x−3.g'(x) = \frac{d}{dx}(1) + \frac{d}{dx}(3x^{-3}) + \frac{d}{dx}(6x^{-2}) = 0 - 9x^{-4} - 12x^{-3}. Next, we find the derivative of the second function, h(x)=−8x2−6h(x) = -8x^{2}-6. The derivative of a constant times a function is the constant times the derivative of the function, and the derivative of a constant is 00.Using the power rule again, we find the derivative of −8x2-8x^{2} as follows: \newlineddx(−8x2)=−8×2×x2−1=−16x.\frac{d}{dx}(-8x^{2}) = -8 \times 2 \times x^{2-1} = -16x. The derivative of −6-6 is 00, so ddx(−6)=0\frac{d}{dx}(-6) = 0.Now we can combine the derivatives of the parts of the second function: \newlineh′(x)=ddx(−8x2)+ddx(−6)=−16x+0=−16x.h'(x) = \frac{d}{dx}(-8x^{2}) + \frac{d}{dx}(-6) = -16x + 0 = -16x. Using the product rule, we find the derivative of the product of g(x)g(x) and h(x)h(x): \newlinef′(x)=g′(x)h(x)+g(x)h′(x).f'(x) = g'(x)h(x) + g(x)h'(x). Substitute the derivatives and original functions into the product rule formula: \newlineh(x)=−8x2−6h(x) = -8x^{2}-600 Now we need to simplify the expression. We will distribute the terms and combine like terms where possible. Let's start with the first part of the product rule: \newlineh(x)=−8x2−6h(x) = -8x^{2}-611 Now let's distribute the second part of the product rule: \newlineh(x)=−8x2−6h(x) = -8x^{2}-622 Combine the distributed terms from both parts of the product rule: \newlineh(x)=−8x2−6h(x) = -8x^{2}-633
  16. Simplify Expression: Now we can combine the derivatives of the parts of the first function: g′(x)=ddx(1)+ddx(3x−3)+ddx(6x−2)=0−9x−4−12x−3.g'(x) = \frac{d}{dx}(1) + \frac{d}{dx}(3x^{-3}) + \frac{d}{dx}(6x^{-2}) = 0 - 9x^{-4} - 12x^{-3}. Next, we find the derivative of the second function, h(x)=−8x2−6h(x) = -8x^{2}-6. The derivative of a constant times a function is the constant times the derivative of the function, and the derivative of a constant is 00. Using the power rule again, we find the derivative of −8x2-8x^{2} as follows: ddx(−8x2)=−8×2×x2−1=−16x\frac{d}{dx}(-8x^{2}) = -8 \times 2 \times x^{2-1} = -16x. The derivative of −6-6 is 00, so ddx(−6)=0\frac{d}{dx}(-6) = 0. Now we can combine the derivatives of the parts of the second function: h′(x)=ddx(−8x2)+ddx(−6)=−16x+0=−16xh'(x) = \frac{d}{dx}(-8x^{2}) + \frac{d}{dx}(-6) = -16x + 0 = -16x. Using the product rule, we find the derivative of the product of g(x)g(x) and h(x)=−8x2−6h(x) = -8x^{2}-600: h(x)=−8x2−6h(x) = -8x^{2}-611. Substitute the derivatives and original functions into the product rule formula: h(x)=−8x2−6h(x) = -8x^{2}-622. Now we need to simplify the expression. We will distribute the terms and combine like terms where possible. Let's start with the first part of the product rule: h(x)=−8x2−6h(x) = -8x^{2}-633. Now let's distribute the second part of the product rule: h(x)=−8x2−6h(x) = -8x^{2}-644. Combine the distributed terms from both parts of the product rule: h(x)=−8x2−6h(x) = -8x^{2}-655. Simplify by combining like terms: h(x)=−8x2−6h(x) = -8x^{2}-666.

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