A calculus problem by Akshat Sharda

Calculus Level 3

d d x sin ⁡ ( 7 x + ln ⁡ ( 5 x ) ) \frac{d}{dx} \sqrt{ \sin (7x+\ln(5x)) }

If the value of above expression is in the form ( A x + 1 ) cos ⁡ ( A x + ln ⁡ ( B x ) ) C x sin ⁡ ( A x + ln ⁡ ( B x ) ) \frac{(Ax+1) \cos (Ax+\ln(Bx))}{Cx\sqrt{ \sin( Ax+\ln(Bx))}}

Find A + B + C A+B+C .


The answer is 14.

Akshat Sharda by Akshat Sharda , 21, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Chew-Seong Cheong
Feb 18, 2016

Let θ = 7 x + ln ⁡ ( 5 x ) ⇒ d θ d x = 7 + 1 x = 7 x + 1 x \theta = 7x + \ln(5x)\quad \Rightarrow \dfrac{d\theta}{dx} = 7 + \dfrac{1}{x} = \dfrac{7x+1}{x}

d d x sin ⁡ ( 7 x + ln ⁡ ( 5 x ) ) = d d x sin ⁡ 1 2 θ = 1 2 sin ⁡ − 1 2 θ cos ⁡ θ d θ d x = ( 7 x + 1 ) cos ⁡ ( 7 x + ln ⁡ ( 5 x ) ) 2 x sin ⁡ ( 7 x + ln ⁡ ( 5 x ) ) \begin{aligned} \frac{d}{dx} \sqrt{\sin(7x+\ln(5x))} & = \frac{d}{dx} \sin^\frac{1}{2} \theta \\ & = \frac{1}{2} \sin^{-\frac{1}{2}} \theta \cos \theta \frac{d\theta}{dx} \\ & = \frac{(7x+1)\cos (7x + \ln(5x))}{2x\sqrt{\sin(7x+\ln(5x))}} \end{aligned}

⇒ A + B + C = 7 + 5 + 2 = 14 \Rightarrow A+B+C = 7+5+2 = \boxed{14}

5 Helpful 0 Interesting 1 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...