Question

How do you find the derivative of $f(x) = \dfrac{1}{4}{x^2} - x + 4$?

Answer 1

The above problem is based on the differentiation of the given function.
Derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument.
We will perform normal differentiation for the function given without applying any rule of differentiation.

Complete step-by-step solution:
Let’s discuss more about differentiation and then we will perform the differentiation of the given function.
The formula used to perform the differentiation is;
$\dfrac{{d{x^n}}}{{dx}} = n{x^{n - 1}}$ (In this x is the given function here with power n, after differentiation power of function x keeps on decreasing by one power less than the given value in original).
We must keep in mind that the derivative of the constant is always zero.
Now, we will perform the derivative of the given function provided in the question.
$\Rightarrow \dfrac{{d\dfrac{1}{4}{x^2} - x + 4}}{{dx}}$(Given function under derivation with respect to x).
$\Rightarrow \dfrac{2}{4}x - 1$ (We may further simplify the function after differentiation power of both the x reduced to 1 and zero respectively)
$\Rightarrow \dfrac{1}{2}x - 1$

$\dfrac{1}{2}x - 1$ is the required answer.

Note: Derivatives has many applications such as finding the critical points in mathematics, intervals of increase and decrease, relative maxima and minima, concavity and inflection points, curve sketching with derivative, optimization using the closed intervals, linear approximation, Used in limits by applying L- hospital rule, Used in many engineering problems such as for converting signals as ramp, unit or parabolic(When a parabolic function is derived, ramp function is obtained and when ramp function is derived unit signal is obtained).