1
MHT CET 2023 10th May Morning Shift
+2
-0

If $$\mathrm{f}(x)=\mathrm{e}^x, \mathrm{~g}(x)=\sin ^{-1} x$$ and $$\mathrm{h}(x)=\mathrm{f}(\mathrm{g}(x))$$, then $$\frac{\mathrm{h}^{\prime}(x)}{\mathrm{h}(x)}$$ is

A
$$\mathrm{e}^{\sin ^{-1} x}$$
B
$$\frac{1}{\sqrt{1-x^2}}$$
C
$$\sin ^{-1} x$$
D
$$\frac{\mathrm{e}^{\sin ^{-1} x}}{\sqrt{1-x^2}}$$
2
MHT CET 2023 9th May Evening Shift
+2
-0

If $$y$$ is a function of $$x$$ and $$\log (x+y)=2 x y$$, then the value of $$y^{\prime}(0)$$ is

A
1
B
$$-$$1
C
2
D
0
3
MHT CET 2023 9th May Evening Shift
+2
-0

If $$x^{\mathrm{k}}+y^{\mathrm{k}}=\mathrm{a}^{\mathrm{k}}(\mathrm{a}, \mathrm{k}>0)$$ and $$\frac{\mathrm{d} y}{\mathrm{~d} x}+\left(\frac{y}{x}\right)^{\frac{1}{3}}=0$$, then $$\mathrm{k}$$ has the value

A
$$\frac{1}{3}$$
B
$$\frac{2}{3}$$
C
$$\frac{1}{4}$$
D
$$\frac{2}{7}$$
4
MHT CET 2023 9th May Evening Shift
+2
-0

If $$\mathrm{g}$$ is the inverse of $$\mathrm{f}$$ and $$\mathrm{f}^{\prime}(x)=\frac{1}{1+x^3}$$, then $$\mathrm{g}^{\prime}(x)$$ is

A
$$\frac{1}{1+(g(x))^3}$$
B
$$1+(g(x))^3$$
C
$$\frac{g(x)}{1+(g(x))^3}$$
D
$$\frac{(\mathrm{g}(x))^3}{1+(\mathrm{g}(x))^3}$$
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