Question: Find the derivative of the function f(x)=x3−5×2+6x−2f(x) = x^3 – 5x^2 + 6x – 2f(x)=x3−5×2+6x−2.

Question: Find the derivative of the function f(x)=x3−5×2+6x−2f(x) = x^3 – 5x^2 + 6x – 2f(x)=x3−5x2+6x−2.

Solution:

To find the derivative of the function f(x)=x3−5×2+6x−2f(x) = x^3 – 5x^2 + 6x – 2f(x)=x3−5x2+6x−2, we will use the power rule of differentiation. The power rule states that if f(x)=xnf(x) = x^nf(x)=xn, then f′(x)=nxn−1f'(x) = nx^{n-1}f′(x)=nxn−1.

Applying the power rule to each term in the function:

1. For the term x3x^3x3: ddx(x3)=3×2\frac{d}{dx}(x^3) = 3x^2dxd​(x3)=3x2
2. For the term −5×2-5x^2−5x2: ddx(−5×2)=−5⋅2x=−10x\frac{d}{dx}(-5x^2) = -5 \cdot 2x = -10xdxd​(−5x2)=−5⋅2x=−10x
3. For the term $6x$: ddx(6x)=6\frac{d}{dx}(6x) = 6dxd​(6x)=6
4. For the constant term $-2$: ddx(−2)=0\frac{d}{dx}(-2) = 0dxd​(−2)=0

Now, combining these results, the derivative of the function $f(x)$ is: f′(x)=3×2−10x+6f'(x) = 3x^2 – 10x + 6f′(x)=3x2−10x+6

So, the derivative of f(x)=x3−5×2+6x−2f(x) = x^3 – 5x^2 + 6x – 2f(x)=x3−5x2+6x−2 is: f′(x)=3×2−10x+6f'(x) = 3x^2 – 10x + 6f′(x)=3x2−10x+6