R2

{
   "voice_prompt": "",
   "manuscript": {
       "title": {
           "text": "How do you find the derivative of polynomial functions?",
           "audio": "How do you find the derivative of polynomial functions?"
       },
       "description": {
           "text": "To find the derivative of a polynomial function, you use the power rule. If f(x) = x^n, then f'(x) = n * x^(n - 1).",
             "latex": "f(x) = x^n \\Rightarrow f'(x) = n \\cdot x^{n - 1}"
           "audio": "To find the derivative of a polynomial function, you use the power rule. If f of x equals x to the power of n, then f prime of x equals n times x to the power of n minus one."
       },
       "scenes": [
           {
               "text": "Let's take a look at this function: f of x equals 4 times x cubed, minus 3 times x squared, plus 5. f of x is a polynomial function, which is a function made up of one or more terms, where each term is a constant multiplied by x raised to a non-negative integer power.",
               "latex": "f(x) = 4x^3 - 3x^2 + 5"
           },
           {
               "text": "Here are some more examples of polynomial functions: g of x equals x to the power of 4, minus x squared, plus 2, and h of x equals 2 times x, minus 3.",
               "latex": "g(x) = x^4 - x^2 + 2,\\quad h(x) = 2x - 3"
           },
           {
               "text": "Now, let\u2019s go over the power rule again. Remember, the power rule states that to find the derivative of x to the power of n, you bring down the exponent as a multiplier and then reduce the exponent by one.",
               "latex": "\\frac{d}{dx}(x^n) = n \\cdot x^{n - 1}"
           },
           {
               "text": "Let's use the power rule to find the derivative of f of x equals x cubed. In this case, n equals 3. So, let's replace n with 3 in the formula, and you get that the derivative of f, f prime equals 3 times x to the power of 3 minus 1, or 3 times x squared.",
               "latex": "f(x) = x^3 \\Rightarrow f'(x) = 3x^2"
           },
           {
               "text": "Now, let's look at another example. What is the derivative of the function g of x equals x? Remember that x can also be written as x to the first power, so in this case, n equals 1. So, let's replace n with 1 in the formula, and you get that the derivative of g, g prime equals 1 times x to the power of 1 minus 1, or simply 1.",
               "latex": "g(x) = x \\Rightarrow g'(x) = 1"
           },
           {
               "text": "But what happens when the function is a constant, like f of x equals 7? Well, you can write f of x as 7 times x to the power of 0, as any number raised to the power of 0 equals 1. Applying the power rule, replacing n by 0, you get that the derivative of f, f prime equals 0 times x to the power of -1, which is 0, as any number multiplied by 0 is 0. So, the derivative of any constant function is 0.",
               "latex": "f(x) = 7 = 7x^0 \\Rightarrow f'(x) = 0 \\cdot x^{-1} = 0"
           },
           {
               "text": "Now, what if you have a coefficient in front of the power of x, like in g of x equals 5 times x cubed? The rule is that the constant multiplier is not affected when taking the derivative. You simply apply the power rule and then multiply by the coefficient. So, the derivative of g, g prime of x in this case equals 5 times 3 times x to the power of 3 minus 1, which is 15 times x squared.",
               "latex": "g(x) = 5x^3 \\Rightarrow g'(x) = 15x^2"
           },
           {
               "text": "Finally, for a polynomial with multiple terms added or subtracted, the derivative of the entire polynomial is simply the sum or difference of the derivatives of each individual term. Let's go back to the function presented to you at the start: f of x equals 4 times x cubed, minus 3 times x squared, plus 5 and apply the power rule to each term: The derivative of 4x cubed is 4 times 3 times x squared, which simplifies to 12 x squared. The derivative of -3 times x squared is -3 times 2 times x, which simplifies to -6 times x. And finally, the derivative of the constant 5 is 0. Therefore, the derivative of f, f prime equals 12 times x squared minus 6 times x.",
               "latex": "f(x) = 4x^3 - 3x^2 + 5 \\Rightarrow f'(x) = 12x^2 - 6x"
           }
       ],
       "outro": {
         "text": "To find the derivative of a polynomial function, you use the power rule. If f(x) = x^n, then f'(x) = n * x^(n - 1).",
           "latex": "f(x) = x^n \\Rightarrow f'(x) = n \\cdot x^{n - 1}"
         "audio": "To find the derivative of a polynomial function, you use the power rule. If f of x equals x to the power of n, then f prime of x equals n times x to the power of n minus one."
       }
   }
}

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