How do you find the derivative of #f(x)=1/4x^2-x+4#?

Question

Charles Anctil · Accepted Answer

#1/2x-1#

To find the derivative of a polynomial, we can use the sum/difference rules for differentiation, which means that we can take the derivative of each term separated by an addition/subtraction sign separately.

First, let's find the derivative of the first term by using the power rule, which states that the derivative of #x^n# is #nx^(n-1)#:

When there's a constant in front of a variable, just put the constant to the side for the moment and focus on differentiating the non-constant variable. After that is done, the constant should be multiplied by the new derivative that is obtained.

#1/4x^2# becomes #1/4(2*x^(2-1))#, which in turn becomes #1/2x#.

Next, let's take the derivative of the second term, which is #-x#, and the derivative of a variable by itself is just 1. Taking into account the negative sign, this turns out to be #-1#.

The last term is a constant, and the derivative of any constant is #0#, so that will replace the 4.

Combining all of our answers together, our final result is #1/2x-1#.

Samantha Adkison · Answer

undefined To find the derivative of $ f(x) = \frac{1}{4}x^2 - x + 4 $, you can apply the power rule and the constant rule. The derivative of $ x^n $ is $ nx^{n-1} $, and the derivative of a constant is 0.

So, for $ f(x) = \frac{1}{4}x^2 - x + 4 $:
1. The derivative of $ \frac{1}{4}x^2 $ is $ \frac{1}{4} \times 2x = \frac{1}{2}x $.
2. The derivative of $ -x $ is $ -1 $.
3. The derivative of the constant term 4 is 0.

Therefore, the derivative of $ f(x) = \frac{1}{4}x^2 - x + 4 $ is $ f'(x) = \frac{1}{2}x - 1 $.

Samantha Adkison · Answer

undefined To find the derivative of the function $ f(x) = \frac{1}{4}x^2 - x + 4 $, we can apply the power rule and the constant multiple rule of differentiation:

1. For $ \frac{1}{4}x^2 $, using the power rule, the derivative is $ \frac{1}{4} \cdot 2x = \frac{1}{2}x $.
2. For $ -x $, the derivative of a constant multiple of x is simply the constant itself. So, the derivative of $ -x $ is $ -1 $.
3. For the constant term 4, the derivative of a constant is zero.

Combining these results, the derivative of the function $ f(x) = \frac{1}{4}x^2 - x + 4 $ is:

$$ f'(x) = \frac{1}{2}x - 1 $$