Q.38 - Calculus

Question

Use the differentiation rules developed in this section to find

the derivatives of the functions

$5x^3-2x^2+7$

Step-by-Step Solution

Answer

The derivative of  the given  function is $15x^2-4x$

Step1. Given information

Given function is $f\left(x\right)=5x^3-2x^2+7$

We need to find out the derivatives of the given function. 

Step2. finding the derivatives of the function

Here the function is $f\left(x\right)=5x^3-2x^2+7$. When we apply the  differentiation rules on the given function , we get

$$\begin{gathered} \frac{d}{d\mathrm{x}}(5{\mathrm{x}}^3-2{\mathrm{x}}^2+7)=\frac{d\left(5{\mathrm{x}}^3\right)}{d\mathrm{x}}-\frac{d\left(2{\mathrm{x}}^2\right)}{d\mathrm{x}} +\frac{d\left(7\right)}{d\mathrm{x}} \\ \mathrm{derivative} \mathrm{of} \mathrm{a} \mathrm{constant} \mathrm{is} 0 , \mathrm{thus} \frac{d\left(7\right)}{d\mathrm{x}}=0 \mathrm{and} \\ \mathrm{the} \mathrm{derivative} \mathrm{of} \frac{d}{d\mathrm{x}}\left({\mathrm{x}}^{\mathrm{n}}\right)={\mathrm{nx}}^{\mathrm{n}-1} ,\mathrm{for} \mathrm{any} \mathrm{non} \mathrm{zero} \mathrm{rational} \mathrm{number} \mathrm{n} \mathrm{by} \mathrm{using} \mathrm{power} \mathrm{rule} \\ \mathrm{hence} \frac{d\left(5{\mathrm{x}}^3\right)}{d\mathrm{x}} =5\frac{d}{d\mathrm{x}}\left({\mathrm{x}}^3\right) \\ =5\times 3\times {\mathrm{x}}^2 \\ =15{\mathrm{x}}^2 \\ \frac{d\left(2{\mathrm{x}}^2\right)}{d\mathrm{x}}=2\frac{d\left({\mathrm{x}}^2\right)}{d\mathrm{x}} \\ =2\times 2\mathrm{x} \\ =4\mathrm{x} \\ \mathrm{thus} \mathrm{the} \mathrm{derivatives} \mathrm{of} \mathrm{t} \mathrm{he} \mathrm{function} ,\frac{d}{d\mathrm{x}}(5{\mathrm{x}}^3-2{\mathrm{x}}^2+7)=15{\mathrm{x}}^2-4\mathrm{x}+0 \\ =15{\mathrm{x}}^2-4\mathrm{x} \end{gathered}$$