38. If
$$
y=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots . .+\frac{x^n}{n!}
$$
show that
$$
\frac{d y}{d x}-y+\frac{x^n}{n!}=0
$$
39. Differentiate the following functions w.r.t. x :
$$
e^{x \log a}+e^{a \log x}+e^{a \log a}
$$
40. Differentiate the following functions w.r.t. x :
$$
\frac{a \cos x+b \sin x+c}{\sin x}
$$
41. Differentiate the following functions w.r.t. x :
$$
a_0 x^n+a_1 x^{n-1}+a_2 x^{n-2}+\ldots . .+a_{n-1} x
$$
42. Find the rate at which the function
$$
f(x)=x^4-2 x^3+3 x^2+x+5
$$
changes with respect to x.
43. If $$y=\frac{2 x^9}{3}-\frac{5}{7} x^7+6 x^3-x$$, find $$\frac{d y}{d x}$$ at $$x=1$$.
44. Find the derivative of the function $$f$$, defined by
$$
\mathrm{f}(\mathrm{x})=\mathrm{mx}+\mathrm{c} \text { at } \quad \mathrm{x}=0 .
$$
45. Differentiate the following functions from first principle
$$
(\mathrm{x}+1)(\mathrm{x}+2)(\mathrm{x}+3)
$$
46. Derivative of a constant multiple of a function
47. Derivative of the Logarithmic Function
48. Differentiate the following functions w.r.t. $$x$$
$$
\frac{x}{3}-\frac{3}{x}+\sqrt{x}-\frac{1}{\sqrt{x}}+x^2-2^x+6^{x^{-2 / 3}}-\frac{2}{3} x^6
$$
49. If for
$$
\mathrm{f}(\mathrm{x})=\lambda \mathrm{x}^2+\mu \mathrm{x}+12, \mathrm{f}^{\prime}(4)=15 \text { and } \mathrm{f}^{\prime}(2)=11,
$$
then find $$\lambda$$ and $$\mu$$
$$
\text { 50. If } y=\left(\sqrt{\frac{x}{a}}+\sqrt{\frac{a}{x}}\right) \text {, show that } 2 x y \frac{d y}{d x}=\left(\frac{x}{a}-\frac{a}{x}\right)
$$
51. $$y=\frac{1-\tan ^2 x / 2}{1+ \tan ^2 x / 2}$$, find $$\frac{d y}{d x}$$
52. Differentiate the following functions w. r. t. $$x$$ :
(i) $$e^x \sin x$$
(ii) $$\mathrm{x}^2 \log _{\mathrm{e}} \mathrm{x}$$
(iii) $$\left(3 x^2+2\right)^2$$
(iv) $$(\mathrm{x}+2)(\mathrm{x}+3)$$