Find the derivative of y = (1/10)x^5 + x^4 + 3x - sqrt(3).

Step-by-step solution

1. 1

   Differentiate the leading term and simplify the coefficient. The power rule gives $\left(\frac{1}{10}x^{5}\right)^{\prime}=\frac{1}{10}\cdot 5x^{4}=\frac{5}{10}x^{4}=\frac12 x^{4}.$ A frequent error here is to write $2x^{4}$, which would come from dividing $10$ by $5$ instead of multiplying $\frac1{10}$ by $5$.

2. 2

   Differentiate the quartic term. $\left(x^{4}\right)^{\prime}=4x^{3}.$

3. 3

   Differentiate the linear term. $\left(3x\right)^{\prime}=3 .$

4. 4

   Recognise the irrational constant. $\sqrt3$ is a fixed number, roughly $1.732$, not a function of $x$, so $\left(-\sqrt3\right)^{\prime}=0$. The presence of a radical does not change this.

5. 5

   Assemble the derivative. $y^{\prime}=\frac12 x^{4}+4x^{3}+3 .$ Note that $y^{\prime}>0$ whenever $x>0$, and at $x=0$ the derivative equals $3$, so the curve is rising at the origin.