Differentiate 3x+2(37,500,000/x) I know the answer is suppose to be 3(x^2-25,000,000)/x^2 but I'm not sure how to get there.
Got it thanks anyways... I for some reason wasn't putting 3 over x^2 to simplify. dumb mistake.
d ( 1 / x ^ n ) / dx = - n * x ^ ( - n - 1 )
d ( 1 / x ) / dx = d ( 1 / x ^ 1 ) / dx = - 1 * x ^ ( - 1 - 1 ) = - 1 * x ^ - 2 = - 1 / x ^ 2
d ( 3x ) / dx = 3
d [ 3 x + 2 ( 37,500,000 / x ) ] / dx =
3 + 2 * 37,500,000 * ( - 1 ) / x ^ 2 =
3 - 75,000,000 / x ^ 2
To differentiate the given expression, we will apply the power rule and the constant multiple rule of differentiation. Here's how you can go step by step to simplify and differentiate the expression:
1. Start with the given expression: 3x + 2(37,500,000/x).
2. Distribute the 2 to the terms inside the parentheses: 3x + (2 * 37,500,000)/x.
This simplifies to: 3x + 75,000,000/x.
3. To differentiate separately, rewrite the expression as: 3x + 75,000,000 * x^(-1).
4. Apply the power rule of differentiation to the first term, which states that d/dx (x^n) = n * x^(n-1). The derivative of 3x is simply 3.
5. For the second term, apply the constant multiple rule where d/dx (c * f(x)) = c * (d/dx f(x)). In this case, the constant (coefficient) is 75,000,000.
Therefore, d/dx (75,000,000 * x^(-1)) = 75,000,000 * d/dx (x^(-1)).
6. Apply the power rule to the second term by differentiating the function x^(-1). Since x^(-1) can be written as 1/x, the derivative is -1 * x^(-2), which simplifies to -x^(-2).
So, d/dx (75,000,000 * x^(-1)) = 75,000,000 * (-x^(-2)).
7. Simplify the second term further: -75,000,000 * x^(-2).
8. Combine the derivatives obtained in steps 4 and 7 to get the final result:
Final derivative = 3 + (-75,000,000 * x^(-2)).
9. To obtain the expression in the desired form, bring the second term over a common denominator (x^2) by multiplying the entire expression by x/x.
Final result = (3x^3 - 75,000,000)/x^2.
Simplifying it further, we get: 3(x^2 - 25,000,000)/x^2.
Therefore, the given expression differentiates to 3(x^2 - 25,000,000)/x^2.