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

Sagot :

We need to find the derivative of the function

[tex]f\mleft(x\mright)=6x^{4}-7x^{3}+2x+\sqrt{2}​[/tex]

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:

[tex](ax^n)^{\prime}=n\cdot a\cdot x^{n-1}[/tex]

Step 1

Find the derivatives of each term:

[tex]\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}[/tex]

Step 2

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

[tex]f^{\prime}(x)=24x^{3}-21x^{2}+2[/tex]

Answer

Therefore, the derivative of the given function is

[tex]24x^3-21x^2+2[/tex]