# Use the theorems on derivatives to find the derivatives of the following function: f(x)=3x^{5}-2x^{4}-5x+7+4x^{-2}

Derivatives
Use the theorems on derivatives to find the derivatives of the following function:
$$\displaystyle{f{{\left({x}\right)}}}={3}{x}^{{{5}}}-{2}{x}^{{{4}}}-{5}{x}+{7}+{4}{x}^{{-{2}}}$$

2021-05-05
Step 1
There are two operations addition and substraction in the given function which is a polynomial. We use sum/difference rule of differentiation, $$\displaystyle{\left({f}\pm{g}\right)}'={f}'\pm{g}'$$
Also we know that the derivative of $$\displaystyle{x}^{{{a}}}\ {i}{s}\ {a}{x}^{{{a}-{1}}}$$.
Derivative of a constant is always zero.
Step 2
Given
$$\displaystyle{f{{\left({x}\right)}}}={3}{x}^{{{5}}}-{2}{x}^{{{4}}}-{5}{x}+{7}+{4}{x}^{{-{2}}}$$
$$\displaystyle{f}'{\left({x}\right)}={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({3}{x}^{{{5}}}\right)}-{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({2}{x}^{{{4}}}\right)}-{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({5}{x}\right)}+{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({7}\right)}+{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({4}{x}^{{-{2}}}\right)}$$
$$\displaystyle={15}{x}^{{{4}}}-{8}{x}^{{{3}}}-{5}+{0}-{8}{x}^{{{3}}}$$
Step 3
Therefore the derivative is given as,
$$\displaystyle{f}'{\left({x}\right)}={15}{x}^{{{4}}}-{8}{x}^{{{3}}}-{\frac{{{8}}}{{{x}^{{{3}}}}}}-{5}$$