# Find all the antiderivatives of the following function. Check your work by taking derivatives. h(z)=-6z^{-7}

Find all the antiderivatives of the following function. Check your work by taking derivatives.
$$\displaystyle{h}{\left({z}\right)}=-{6}{z}^{{-{7}}}$$

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Nathalie Redfern
Step 1
To find the antiderivative, we will integrate.
$$\displaystyle\int{h}{\left({z}\right)}{\left.{d}{z}\right.}$$
$$\displaystyle=\int-{6}{z}^{{-{7}}}{\left.{d}{z}\right.}$$
$$\displaystyle=-{6}\int{z}^{{-{7}}}{\left.{d}{z}\right.}$$
$$\displaystyle=-{6}{\frac{{{z}^{{-{7}+{1}}}}}{{-{7}+{1}}}}+{C}$$
$$\displaystyle=-{6}{\frac{{{z}^{{-{6}}}}}{{-{6}}}}+{C}$$
$$\displaystyle{z}^{{-{6}}}+{C}$$
$$\displaystyle={\frac{{{1}}}{{{z}^{{{6}}}}}}+{C}$$
C=integrating constant
Step 2
Then we check it using derivative.
$$\displaystyle{\frac{{{d}}}{{{\left.{d}{z}\right.}}}}{\left({\frac{{{1}}}{{{z}^{{{6}}}}}}+{C}\right)}$$
$$\displaystyle={\frac{{{d}}}{{{\left.{d}{z}\right.}}}}{\left({z}^{{-{6}}}+{C}\right)}$$
$$\displaystyle=-{6}{z}^{{-{6}+{1}}}+{0}$$
$$\displaystyle=-{6}{z}^{{-{7}}}$$
So it shows that the answer is correct.
Answer: $$\displaystyle{\frac{{{1}}}{{{z}^{{{6}}}}}}+{C}$$