Question

attempt 1: 10 attempts remaining. use the rules of derivatives to calculate the derivative of the following function and simplify if possible. (h(x)=-\frac{1}{5}x^{25}+sqrt{x}) (h(x)=)

Explanation:

Step1: Recall power - rule for derivatives

The power - rule states that if $y = ax^n$, then $y^\prime=anx^{n - 1}$. For the function $h(x)=-\frac{1}{5}x^{25}+\sqrt{x}=-\frac{1}{5}x^{25}+x^{\frac{1}{2}}$.

Step2: Differentiate the first term

For $y_1 = -\frac{1}{5}x^{25}$, using the power - rule with $a =-\frac{1}{5}$ and $n = 25$, we have $y_1^\prime=-\frac{1}{5}\times25x^{25 - 1}=- 5x^{24}$.

Step3: Differentiate the second term

For $y_2=x^{\frac{1}{2}}$, using the power - rule with $a = 1$ and $n=\frac{1}{2}$, we get $y_2^\prime=\frac{1}{2}x^{\frac{1}{2}-1}=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}}$.

Step4: Combine the derivatives

Since $h(x)=y_1 + y_2$, then $h^\prime(x)=y_1^\prime + y_2^\prime$. So $h^\prime(x)=-5x^{24}+\frac{1}{2\sqrt{x}}$.

Answer:

$-5x^{24}+\frac{1}{2\sqrt{x}}$