Question

How do you find the derivative of `f(x)=-2x^-3+x^2-7`?

Answer 1

`f'(x)=6x^-4+2x`

Explanation:

Use the power rule for differentiation:

We know that the derivative of `x^n` is `nx^(n-1)`.

The function `f(x)=-2x^(-3)+x^2-7` has three parts.

We apply the power rule on each part.

`color(red)(Part-1`: `-2x^-3`
Derivative: `-2(-3)x^((-3-1))`
Derivative: `color(red)(6x^-4`

`color(blue)(Part-2`: `x^2`
Derivative: `(2)x^((2-1))`
Derivative: `color(blue)(2x`

`color(green)(Part-3`: `7`
The derivative of a constant is zero, so:
Derivative: `color(green)(0`

We add the derivatives of each part up:

`color(red)(6x^-4)+color(blue)(2x)+color(green)(0)`

So the derivative would be:

`f'(x)=6x^-4+2x`

Answer 2

`6x^(-4)+2x`

Explanation:

Since we're dealing with a polynomial, we can find the derivative with the help of the Power Rule.

We simply multiply the exponent by the coefficient, and decrement the power by one. Recall that the derivative of a constant is zero.

We now have:

`6x^(-4)+2x`

Hope this helps!