Question:

The derivative of eaxcosbxe^{ax} \cos\,bx with respect to xx is reaxre^{ax} cos(bxTan1ba)\cos\, (bx-Tan^{-1} \frac {b}{a}) When a>0,b>0a >0, b > 0,the value of rr is ............

Updated On: May 19, 2024
  • abab
  • a+ba+b
  • a2+b2\sqrt {a^2+b^2}
  • 1ab\frac {1} {\sqrt {ab}}
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The Correct Option is C

Solution and Explanation

Given, ddx(eaxcosbx)=reaxcos(bx+α)\frac{d}{d x}\left(e^{a x} \cos b x\right)=r e^{a x} \cos (b x+\alpha), then
r=?r=?
Let y=eaxcosbxy=e^{a x} \cdot \cos b x
Let y=eaxcosbxy=e^{a x} \cdot \cos b x
dydx=aeaxcosbxbeaxsinbx\frac{d y}{d x}=a e^{a x} \cdot \cos b x-b e^{a x} \cdot \sin b x
dydx=eax(acosbxbsinbx)\frac{d y}{d x}=e^{a x}(a \cos b x-b \sin b x)
$a=r \cos \alpha
b=r \sin \alpha\}$...(i)
Then, dydx=eaxr{cosbxcosαsinbxsinα}\frac{d y}{d x} =e^{a x} \cdot r\{\cos b x \cdot \cos \alpha-\sin b x \cdot \sin \alpha\}
dydx=eaxrcos(bx+α)\frac{d y}{d x}=e^{a x} \cdot r \cos (b x+\alpha)...(ii)
Where, tanα=baα=tan1(ba)\tan \alpha=\frac{b}{a} \Rightarrow \alpha=\tan ^{-1}\left(\frac{b}{a}\right)
and r2(cos2α+sin2α)=a2+b2r^{2}\left(\cos ^{2} \alpha+\sin ^{2} \alpha\right)=a^{2}+b^{2}
r2=a2+b2r^{2}=a^{2}+b^{2}
r=a2+b2r=\sqrt{a^{2}+b^{2}}
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Concepts Used:

Second-Order Derivative

The Second-Order Derivative is the derivative of the first-order derivative of the stated (given) function. For instance, acceleration is the second-order derivative of the distance covered with regard to time and tells us the rate of change of velocity. 

As well as the first-order derivative tells us about the slope of the tangent line to the graph of the given function, the second-order derivative explains the shape of the graph and its concavity. 

The second-order derivative is shown using f’’(x) or d2ydx2f’’(x)\text{ or }\frac{d^2y}{dx^2}.