Find y' for y = 1/5 x^5 - 2/3 x³ + 4x - sqrt(10)
Find $y'$ for $y = \dfrac{1}{5}x^{5} - \dfrac{2}{3}x^{3} + 4x - \sqrt{10}$.
Explanation of Method
Differentiate a polynomial term by term.
Step 1: Differentiate term by term
Apply the sum–difference rule together with the power rule $(x^{n})' = nx^{n-1}$, remembering that $\sqrt{10}$ is a constant: $y' = \frac{1}{5}\cdot 5x^{4} - \frac{2}{3}\cdot 3x^{2} + 4\cdot 1 - 0$.
Step 2: Simplify
The fractional coefficients cancel against the exponents brought down by the power rule: $y' = x^{4} - 2x^{2} + 4$.
Step 3: Verification
\[x = 1:\quad y'(1) = 1 - 2 + 4 = 3\]
\[\Delta x = 10^{-5}:\quad \frac{y(1+10^{-5})-y(1)}{10^{-5}} \approx 3.00001 \approx 3 \;\;\checkmark\]
Result
Final answer:
\[\boxed{\,y' = x^{4} - 2x^{2} + 4\,}\]
Remark
Remark. The coefficients $\tfrac15$ and $\tfrac23$ were chosen so that the derivative comes out tidy. Read backwards, the same relation says that $y$ is an antiderivative of $x^{4}-2x^{2}+4$.