CUET UG Mathematics Applied Differentiation Previous Year Questions (PYQs)
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The second order derivative of which of the following functions is $5^x$?
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Solution
If $y=\left(\frac{1}{x}\right)^x$, then value of $e^e\left(\frac{d^2 y}{dx^2}\right)_{x=e}$ is:
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Solution
Given
$y=\left(\frac{1}{x}\right)^x$
Rewrite:
$y=x^{-x}$
Taking log:
$\ln y=-x\ln x$
Differentiate:
$\frac{1}{y}\frac{dy}{dx}=-(\ln x+1)$
So,
$\frac{dy}{dx}=-x^{-x}(\ln x+1)$
Now differentiate again:
$\frac{d^2y}{dx^2}=x^{-x}\left[(\ln x+1)^2-\frac{1}{x}\right]$
Now put $x=e$
Since $\ln e=1$
$(\ln x+1)^2=(1+1)^2=4$
So,
$\frac{d^2y}{dx^2}\Big|_{x=e}=e^{-e}\left(4-\frac{1}{e}\right)$
Multiply by $e^e$:
$e^e\left(\frac{d^2y}{dx^2}\right)_{x=e}=4-\frac{1}{e}$
If $y=e^{nx}$, then $n^{\text{th}}$ derivative of $y$ is:
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Solution
Given
$y=e^{nx}$
First derivative:
$\frac{dy}{dx}=n e^{nx}$
Second derivative:
$\frac{d^2y}{dx^2}=n^2 e^{nx}$
Continuing this pattern,
$n^{\text{th}}$ derivative:
$\frac{d^ny}{dx^n}=n^n e^{nx}$
Since $y=e^{nx}$,
$\frac{d^ny}{dx^n}=n^n y$
If $y=\log(\sec e^{x^2})$, then $\frac{dy}{dx}=$
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Solution
$y=\log(\sec(e^{x^2}))$
Using property:
$\frac{d}{dx}\log(\sec u)=\tan u\cdot\frac{du}{dx}$
Let $u=e^{x^2}$
Then,
$\frac{du}{dx}=2xe^{x^2}$
Hence,
$\frac{dy}{dx}=2xe^{x^2}\tan(e^{x^2})$
If $t=e^{2x}$ and $y=ln(t^2)$, then $\frac{d^2y}{dx^2}$ is
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Solution
CUET UG Mathematics Applied
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CUET UG Mathematics Applied
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and More.
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