1. Differentiate the function with respect to $$\mathrm{x}, \log _x x$$.
2. Differentiate the function with respect to $$\mathrm{x}, e^{3 \log x}$$.
3. Differentiate the function with respect to $$\mathrm{x}, 2^{\log _2 x}$$.
4. Differentiate the function with respect to $$\mathrm{x}, 5\left(2^{3 \log _2 x}\right)$$
5. Differentiate the function with respect to $$\mathrm{x}, 5 e^x$$.
6. Differentiate the function with respect to $$\mathrm{x}, 9 .\left(3^x\right)$$.
7. If $$f(x)=\alpha x^n$$, prove that $$\alpha=\frac{f^{\prime}(1)}{n}$$.
8. If $$f(x)=x^n$$ and if $$f^{\prime}(1)=10$$, find the value of $$n$$.
9. Derivative of a Constant function $$f(x)=c$$
10. Differentiate the following functions w.r.t.x :
(i) $$x^5$$
(ii) $$\frac{1}{x^3}$$
(iv) $$3 x^4$$
(v) $$a x^{-5}$$
(vi) $$7 e^x$$
(vii) $$6 \log _{10} \mathrm{x}$$
(viii) $$2 \log _{\mathrm{e}} x$$
11. Differentiate the following functions w. r. t. $$x$$ : $$\sqrt{x^7}$$
12. Find $$\frac{\mathrm{dy}}{\mathrm{dx}}$$, iif $$\mathrm{y}=\mathrm{x}^2+\sin \mathrm{x}$$.
13. Differentiate the following functions w. r. t. $$x$$ : $$\sqrt{x^7}$$
14. Differentiate the following functions w. r. t. $$x$$ :
$$x^7+e^x$$
15. Find the d.c. of $$3 x^{10}$$ w.r.t. to $$x$$.
16. If $$\mathrm{y}=2 \tan \mathrm{x}$$, find $$\frac{\mathrm{dy}}{\mathrm{dx}}$$.
17. If
$$y=\left(\frac{x^m}{x^n}\right)^{m+n} \cdot\left(\frac{x^n}{x^r}\right)^{n+r} \cdot\left(\frac{x^r}{x^m}\right)^{r+m} \text {, find } \frac{d y}{d x}$$
18. If $$\mathrm{y}=\mathrm{ax}^2+\mathrm{bx}+\mathrm{c}$$, find $$\frac{\mathrm{dy}}{\mathrm{dx}}$$.