CUET Mathematics 2024 - The second order derivative of which of the following functions is 5^x ? | PYQs + Solutions | AfterBoards
Skip to main contentSkip to question navigationSkip to solution
IPMAT Indore Free Mocks Topic Tests

CUET Mathematics 2024 PYQs

View all CUET PYQs →
Collection of AfterBoards IPMAT JIPMAT Past Year Papers with Solutions

CUET Mathematics 2024

Calculus
>
Continuity & Differentiability

Easy

The second order derivative of which of the following functions is 5x5^{\mathrm{x}} ?

✅ Correct Option: 4
The derivative of axa^{x} is given by: \newline ddxax=axln⁡a\frac{d}{dx}a^{x}=a^{x}\ln{a}
Consider option 11. \newline y′=log⁡e5⋅5xlog⁡e5=(log⁡e5)2⋅5xy^{\prime}=\log _{e} 5 \cdot 5^{x} \log _{e} 5=\left(\log _{e} 5\right)^{2} \cdot 5^{x} \newline y′′=(log⁡e5)2⋅5xlog⁡e5=(log⁡e5)3⋅5xy^{\prime \prime}=\left(\log _{e} 5\right)^{2} \cdot 5^{x} \log _{e} 5=\left(\log _{e} 5\right)^{3} \cdot 5^{x}
Consider option 22. \newline y′=1(log⁡e5)2⋅5x⋅log⁡e5=5xlog⁡e5y^{\prime}=\frac{1}{(\log_{e}5)^{2}}\cdot 5^{x}\cdot \log_{e}{5}=\frac{5^{x}}{\log_{e}{5}} \newline y′′=1log⁡e5⋅5x⋅log⁡e5=5xy^{\prime\prime}=\frac{1}{\log_{e}{5}}\cdot 5^{x}\cdot \log_{e}{5}=\boxed{5^{x}}
Consider option 33. \newline y′=1log⁡e5⋅5x⋅log⁡e5=5xy^{\prime}=\frac{1}{\log_{e}{5}}\cdot 5^{x}\cdot \log_{e}{5}=5^{x} \newline y′′=5xlog⁡e5y^{\prime\prime}=5^{x}\log_{e}{5}
Consider option 44. \newline y′=(log⁡e5)2⋅5x⋅log⁡e5=5x(log⁡e5)3y^{\prime}=(\log_{e}{5})^{2}\cdot 5^{x}\cdot \log_{e}{5}=5^{x}(\log_{e}{5})^{3} \newline y′′=(log⁡e5)3⋅5x⋅log⁡e5=5x(log⁡e5)4y^{\prime\prime}=(\log_{e}{5})^{3}\cdot 5^{x}\cdot \log_{e}{5}=5^{x}(\log_{e}{5})^{4}
Hence, option 2 is correct. You can also try integrating it twice too!

Related questions:

CUET Mathematics 2024

If f(x)f(x), defined by f(x)={kx+1 if x≤πcos⁡x if x>πf(x)=\left\{\begin{array}{lll}k x+1 & \text { if } & x \leq \pi \\ \cos x & \text { if } & x>\pi\end{array}\right. is continuous at x=πx=\pi, then the value of kk is :

CUET Mathematics 2024

Let [x][x] denote the greatest integer function. Then match List-I with List-II:

\newline\newline\newline\newline\newline\newline\newline\newline\newline\newline\newline\newline\newline\newline\newline\newline\newline\newline\newline\newline\newline\newline\newline\newline\newline
List-I List-II
(A) ∣x−1∣+∣x−2∣ |x - 1| + |x - 2| (I) is differentiable everywhere except at x=0 x = 0
(B) x−∣x∣ x - |x| (II) is continuous everywhere
(C) x−[x] x - [x] (III) is not differentiable at x=1 x = 1
(D) x ∣x∣ x \, |x| (IV) is differentiable at x=1 x = 1

CUET Mathematics 2024

If t=e2xt=e^{2 x} and y=log⁡et2y=\log _{e} t^{2}, then d2ydx2\frac{d^{2} y}{d x^{2}} is :

Keyboard Shortcuts

  • Left arrow: Previous question
  • Right arrow: Next question
  • S key: Jump to solution
  • Q key: Jump to question