Differentiate y = 8x6 - 3x to the minus 4, plus 4 over x, minus 9.

Step-by-step solution

1. 1

   Rewrite every term as a power of $x$. The power rule only applies to the form $cx^n$, so convert the fraction:

   $\frac4x=4x^{-1}$

   The expression becomes $y=8x^6-3x^{-4}+4x^{-1}-9$. The constant $-9$ is $-9x^0$ and will differentiate to zero.

2. 2

   Differentiate the positive power. With $\frac{d}{dx}x^n=nx^{n-1}$,

   $\frac{d}{dx}\left(8x^6\right)=8\cdot 6x^5=48x^5$

3. 3

   Differentiate the $-3x^{-4}$ term — two negatives meet here.

   $\frac{d}{dx}\left(-3x^{-4}\right)=-3\cdot(-4)x^{-5}=+12x^{-5}$

   The exponent still decreases: $-4-1=-5$, not $-3$. And because the coefficient $-3$ multiplies the negative exponent $-4$, the resulting term is positive.

4. 4

   Differentiate $4x^{-1}$ and the constant.

   $\frac{d}{dx}\left(4x^{-1}\right)=4\cdot(-1)x^{-2}=-4x^{-2},\qquad \frac{d}{dx}(-9)=0$

5. 5

   Combine and present with positive exponents.

   $\frac{dy}{dx}=48x^5+12x^{-5}-4x^{-2}=48x^5+\frac{12}{x^5}-\frac{4}{x^2}$

6. 6

   Spot-check numerically at $x=1$. The formula gives $48+12-4=56$. A central difference on the original function with $h=10^{-4}$ gives $56.00$ as well, confirming both the coefficients and the signs. Note the derivative is undefined at $x=0$, exactly like the original function.