Example 3: Power Rule Derivative

Differentiate $f\left(x\right)= x^3+{5x}^2–7x+3$ with respect to x.

Step 1: Simplify and look for algebraic rewrites.

Here the equation is a simplified as we need it, and there are no algebra rewrites.

You always have to check though.

$f\left(x\right)= x^3+{5x}^2–7x+3$

Step 2: Break the problem down into bitesize chunks based upon the + and – , and identify the derivative rule for each chunk.

Here we have 4 bitesize chunks.

–          The first chunk will be, $x^3$, a power rule .

–          The second chunk will be, ${5x}^2$, a power rule .

–          The third chunk will be, -7x , a power rule .

–          The fourth chunk, 3 , is a constant .

$f\left(x\right)= x^3+{5x}^2–7x+3$

Step 3: Apply derivative rules to each of the chunks you identified in Step 2.

–          We apply the power rule recipe, Bring the power down — Subtract 1 from the power, to the first, second, and third chunks.

–          We apply the constant recipe, the derivative of a constant is always zero, to the second chunk, 9 .

To simplify remember that $x^0=1$.
Anything to the zero power equals 1.

$f\left(x\right)= x^3+{5x}^2–{7x}^1+3$

$f\prime \left(x\right)= 3x^{3–1}+{2\bullet 5x}^{2–1}–1\bullet 7x^{1–1}+0$

$f\prime \left(x\right)= 3x^2+{10x}^1–7x^0+0$

$f\prime \left(x\right)= {3x}^2+10x–7\left(1\right)+0$

$f\prime \left(x\right)= {3x}^2+10x–7$

Final Result:

The derivative of $f\left(x\right)= x^3+{5x}^2–7x+3$  is $f\prime \left(x\right)= {3x}^2+10x–7$.

Meaning:

–          The equation for finding the slope of any tangent line at any x-value of $f\left(x\right)= x^3+{5x}^2–7x+3$  is $f\prime \left(x\right)= {3x}^2+10x–7$.

–          The instantaneous rate of change for every x-value of $f\left(x\right)= x^3+{5x}^2–7x+3$, is found by using the derivative equation, $f\prime \left(x\right)= {3x}^2+10x–7$.

–          The slope of $f\left(x\right)= x^3+{5x}^2–7x+3$at any single x-value can be found by plugging it into the derivative, $f\prime \left(x\right)= {3x}^2+10x–7$.