If u=e^ a x b x and v=e^ a x b x , then what is u (d u)/(d x)+v (d v)/(d x) equal to?

Correct answer: A. a e^ 2 a x aligned & u (d u)/(d x)+v (d v)/(d x)=e^ a x b x[a e^ a x b x+b e^ a x b x] & +e^ a x b x[a e^ a x b x-b e^ a x b x] & =e^ 2 a x [a ^2 b x+b b x b x+. & .a ^2 b x-b b x b x] & =a e^ 2 a x aligned

NDA 2018

Exam: NDA
Year: 2018
Subject: Mathematics
Chapter: Differentiation

NDA 2018 · Mathematics · Differentiation

If u=e^ a x b x and v=e^ a x b x , then what is u (d u)/(d x)+v (d...

u=e^{a x} \sin b x

and

v=e^{a x} \cos b x

, then what is

u \frac{d u}{d x}+v \frac{d v}{d x}

equal to?

Options

A $a e^{2 a x}$
B $\left(a^2+b^2\right) e^{a x}$
C $a b \mathrme^{2 \mathrm{ax}}$
D $(a+b) e^{a x}$

Correct answer

A. $a e^{2 a x}$

Step-by-step solution

\begin{aligned} & u \frac{d u}{d x}+v \frac{d v}{d x}=e^{a x} \sin b x\left[a e^{a x} \sin b x+b e^{a x} \cos b x\right] \\ & +e^{a x} \cos b x\left[a e^{a x} \cos b x-b e^{a x} \sin b x\right] \\ & =e^{2 a x}\left[a \sin ^2 b x+b \sin b x \cos b x+\right. \\ & \left.a \cos ^2 b x-b \sin b x \cos b x\right] \\ & =a e^{2 a x} \end{aligned}

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