S1

{
   "voice_prompt": "",
   "manuscript": {
       "title": {
           "text": "How can you diffrentiate polynominal functions?",
           "audio": "How can you diffrentiate polynominal functions?"
       },
       "description": {
           "text": "
           To differentiate a polynomial function, you can apply the power rule to each term individually.  f(x)=x^n. f'(x)=nx^(n-1).",
           "audio": "To differentiate a polynomial function, you can apply the power rule to each term individually. The power rule says: If f(x) equals x raised to the power of n, then the derivative f'(x) equals n times x raised to the power of (n minus 1)"
       },
       "scenes": [
            }
           {
               "text": "Let's differentiate this polynomial function:f of x equals 4x to the power of 3 minus 5x to the power of 2 plus 6x minus 7.",
               "latex": "f(x) = 4x^3 - 5x^2 + 6x - 7"
           },
           {
               "text": "The power rule states that the derivative of x raised to the power of n is n times x raised to the power of n minus one. Drop the exponent, multiply, then reduce the exponent by one!",
               "latex": "\\f(x)= x^n. f'(x)=n x^{n-1}"
           },
           {
               "text": "Applying the power rule to each term. Start by the first term. The exponent 3 in front of the number 4. Multiply, 4 times 3 equals 12. Then reduce the exponent by 1, 3 minus 1 is 2. So the derivative of this term is: 12x to the power of 2.  ",
               "latex": "4 \\times 3x^{3-1} = 12x^2"
           },
           {
               "text": "Continue with the Second term. Drop the exponent 2 in front of minus 5. Multiply: 2 times minus 5 equals minus 10. Reduce the exponent: 2 minus 1 equals 1.So the derivative becomes: minus 10x. It is not necessary to write the exponent when it's just 1.",
               "latex": ""2 \\times -5x^{2-1} = -10x"
           },
           {
               "text": "Then move on to the next term, 6x. Recall 6x is the same as 6 times x to the power of 1. So you drop the exponent 1 in front of 6. Multiply: 1 times 6 equals 6. Reduce the expoenent, 1 minus 1 equals 0. X to the power of 0 is equal to 1. So you are left with just 6.
               "latex": "1 \\times 6x^{1-1} = 6"
           },
           {
               "text": "Finally, the derivative of a constant, like \\( -7 \\), is zero. This is because constants do not change — their rate of change is zero.",
               "latex": "\\frac{d}{dx}[-7] = 0"
           },
           {
               "text": "And you are done! The derivative of f of x equals four x to the third minus five x squared plus six x minus seven is f prime of x, equals twelve x squared minus ten x plus six.",
               "latex": "f'(x) = 12x^2 - 10x + 6"
           },
           {
               "text": "Knowing how to diffrentiate polynominal functions is useful because it tells you where a function grows fast or slow down. This is very useful in areas like physics were it is discussed how fast a rocket is travelling and in economics, to find out how much it costs to produce phones.",
               "latex": ""
           }
       ],
       "outro": {
           "text": "To differentiate a polynomial function, you can apply the power rule to each term individually.  f(x)=x^n. f'(x)=nx^(n-1)",
           "audio": "To differentiate a polynomial function, you can apply the power rule to each term individually. The power rule says: If f(x) equals x raised to the power of n, then the derivative f'(x) equals n times x raised to the power of (n minus 1)."
       }
   }
}

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