Question

3-34 Differentiate the function.

5. \(f\left( x \right) = {x^{75}} - x + 3\)

Short answer

The differentiation of the function \(f\left( x \right) = {x^{75}} - x + 3\) is \(f'\left( x \right) = 75{x^{74}} - 1\).

Step-by-step solution

Write the formula of the sum and the difference rule, derivative of a constant function, the power rule and the power functions

The Sum and the Difference Rule: \(\frac{d}{{dx}}\left( {f\left( x \right) + g\left( x \right)} \right) = \frac{d}{{dx}}f\left( x \right) + \frac{d}{{dx}}g\left( x \right)\), \(\frac{d}{{dx}}\left( {f\left( x \right) - g\left( x \right)} \right) = \frac{d}{{dx}}f\left( x \right) - \frac{d}{{dx}}g\left( x \right)\)

Derivative of a Constant Function:\(\frac{d}{{dx}}\left( c \right) = 0\)

The Power Rule: \(\frac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\)where \(n\) is a positive integer

The Power function: \(\frac{d}{{dx}}\left( x \right) = 1\)

Find the differentiation of the function

Consider the function \(f\left( x \right) = {x^{75}} - x + 3\). Differentiate the function w.r.t \(x\) by using the sum and the difference rule, derivative of a constant function, the power rule and the power functions.

\(\begin{aligned}\frac{{d\left( {f\left( x \right)} \right)}}{{dx}} &= \frac{{d\left( {{x^{75}} - x + 3} \right)}}{{dx}}\\ &= \frac{d}{{dx}}\left( {{x^{75}}} \right) + \frac{d}{{dx}}\left( { - x} \right) + \frac{d}{{dx}}\left( 3 \right)\\ &= \frac{d}{{dx}}\left( {75{x^{75 - 1}}} \right) - 1\frac{d}{{dx}}\left( x \right) + \frac{d}{{dx}}\left( 3 \right)\\ &= 75{x^{74}} - 1\left( 1 \right) + 0\\ &= 75{x^{74}} - 1\end{aligned}\)

Thus, the derivative of the function \(f\left( x \right) = {x^{75}} - x + 3\) is \(f'\left( x \right) = 75{x^{74}} - 1\).