Question

Find the derivative :f(x) = 6x⁴ -7x³ + 2x + √2

Answer 1

We need to find the derivative of the function

`f\mleft(x\mright)=6x^(4)-7x^(3)+2x+√(2)​`

The derivative of a polynomial equals the sum of the derivatives of each of its terms.

And the derivative of each term axⁿ, where a is the constant multiplying the nth power of x, is given by:

`(ax^n)^(\prime)=n\cdot a\cdot x^(n-1)`

Step 1

Find the derivatives of each term:

`\begin{gathered} (6x^4)^(\prime)=4\cdot6\cdot x^(4-1)=24x^(3) \\ \\ (-7x^3)^(\prime)=3\cdot(-7)\cdot x^(3-1)=-21x^(2) \\ \\ (2x)^(\prime)=1\cdot2\cdot x^(1-1)=2x^0=2\cdot1=2 \\ \\ (\sqrt[]{2})^(\prime)=0,\text{ (since this term doesn't depend on x, its derivative is 0)} \end{gathered}`

Step 2

Add the previous results to find the derivative of f(x):

`f^(\prime)(x)=24x^(3)-21x^(2)+2`

Answer

Therefore, the derivative of the given function is

`24x^3-21x^2+2`