(a) Differentiate ((5^x)/(x^5)) with respect to x.

Let, $\displaystyle y=\frac{5^{\mathrm{x}}}{\mathrm{x}^{5}}=5^{\mathrm{x}} \cdot \mathrm{x}^{-5} \Rightarrow \frac{d y}{d \mathrm{x}}=\left(5^{\mathrm{x}}\right)^{\prime} \cdot \mathrm{x}^{-5}+5^{\mathrm{x}} \cdot\left(\mathrm{x}^{-5}\right)^{\prime}$

\[=\frac{5^{\mathrm{x}}}{\mathrm{x}^{5}} \log 5-\frac{5^{\mathrm{x}+1}}{\mathrm{x}^{6}} \]

Differentiating $\displaystyle -2 \mathrm{x}^{2}-5 \mathrm{x} y+y^{3}=76$, with respect to ' $\displaystyle \mathrm{x}$ '

\[-4 \mathrm{x}-5 y-5 \mathrm{x} \frac{d y}{d \mathrm{x}}+3 y^{2} \frac{d y}{d \mathrm{x}}=0 \]

\[\Rightarrow \frac{d y}{d \mathrm{x}}=\frac{4 \mathrm{x}+5 y}{3 y^{2}-5 \mathrm{x}} \]