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Application Of Derivative Of Trigonometric Functions

Learn equation of tangent line to a curve involving trigonometric function at a point on it. Practice horizontal tangent for trigonometric curves and finding higher order derivatives of sin x by observing a Pattern.

Equation of Tangent Line to a Curve Involving Trigonometric Function

The equation of tangent to the curve \(y=f(x)\) at any point \((a,f(a))\) on it, is given by

\(y-f(a)=f'(a)(x-a)\)

where \(f'(a)\) is the derivative of function \(f\) at \(x=a\) and represents then slope of the tangent.

For example, consider \(y=sin\,x\)

 

The equation of tangent at \(\left ( \dfrac {\pi}{4},\dfrac {1}{\sqrt 2} \right)\) will be

\(y-\dfrac {1}{\sqrt 2}=cos\dfrac {\pi}{4} \left ( x-\dfrac {\pi}{4} \right)\)

\(y-\dfrac {1}{\sqrt 2}=\dfrac {1}{\sqrt 2} \left ( x-\dfrac {\pi}{4} \right)\)

\(\Rightarrow \sqrt 2\,y-1=x-\dfrac {\pi}{4}\)

\(\Rightarrow x-\sqrt 2 y+1-\dfrac {\pi}{4}=0\)

Illustration Questions

Find the equation of tangent line to the curve \(y=(2+x^2)\;(cos\,x)\) at the point  \((0, 2)\)  on it.

A \(y=2\)

B \(3x+2y+7=0\)

C \(2x+y-1=0\)

D \(x=18\)

×

\(y=(2+x^2)\,cos\,x\)

\(\Rightarrow \dfrac {dy}{dx}=(2+x^2)\,\dfrac {d}{dx}\,(cos\,x)+(cos\,x)\dfrac {d}{dx}(2+x^2)\)

\(=(2+x^2)×(-sin\,x)+cos\,x×2x\)

\(\dfrac {dy}{dx}\Bigg |_{x=0}=(2+0)×(-sin\,0)+cos\,0×2×0=0\)

Equation of tangent is 

\(y-y_1=m(x-x_1)\)

\(\Rightarrow\,y-2=0(x-0)\)

\( \Rightarrow y=2\)

Find the equation of tangent line to the curve \(y=(2+x^2)\;(cos\,x)\) at the point  \((0, 2)\)  on it.

A

\(y=2\)

.

B

\(3x+2y+7=0\)

C

\(2x+y-1=0\)

D

\(x=18\)

Option A is Correct

Horizontal Tangent for Trigonometric Curves

To find the point at which tangent is horizontal we solve the equation

\(\dfrac {dy}{dx}=0\)  i.e. Slope = 0

(Horizontal lines have slope 0 )

Illustration Questions

Find the point at which tangent to the curve  \(y=(2\,sinx-x) \left ( \dfrac {-\pi}{2}\leq x\leq \dfrac {\pi}{2} \right)\) is horizontal.

A \(\left ( \dfrac {\pi}{3},\;\sqrt 3 - \dfrac {\pi}{3} \right)\)

B \(\left ( \dfrac {\pi}{6},\;\pi-\sqrt 3 \right)\)

C \((1,-2)\)

D \((\pi,6)\)

×

For horizontal tangent, \(\dfrac {dy}{dx}=0\)

\(\Rightarrow \dfrac {d}{dx}(2\,sin\,x-x)=0\)

\(\Rightarrow 2\,cos\,x-1=0 \)

\(\Rightarrow cos\,x=\dfrac {1}{2}\)

\(\Rightarrow x=\dfrac {\pi}{3}\)

\(\therefore\) Required point     \( \left (\dfrac {\pi}{3},\;\underbrace {\sqrt 3 - \dfrac {\pi}{3}}_{y(\pi/3)} \right)\)

Find the point at which tangent to the curve  \(y=(2\,sinx-x) \left ( \dfrac {-\pi}{2}\leq x\leq \dfrac {\pi}{2} \right)\) is horizontal.

A

\(\left ( \dfrac {\pi}{3},\;\sqrt 3 - \dfrac {\pi}{3} \right)\)

.

B

\(\left ( \dfrac {\pi}{6},\;\pi-\sqrt 3 \right)\)

C

\((1,-2)\)

D

\((\pi,6)\)

Option A is Correct

Finding Higher Order Derivatives of sin x by observing a Pattern

Observe that:

\(\dfrac {d}{dx}(sin\,x)=cos\,x\Rightarrow\dfrac {d^2}{dx^2}(sinx)=-sin\,x\Rightarrow\dfrac {d^3y}{dx^3}=-cos\,x\)

\(\Rightarrow\dfrac {d^4y}{dx^4}=sin\,x\rightarrow\) This pattern will be repeated.

If \(y=sin\,x\)

\(\therefore\)  \(\dfrac {d^ny}{dx^n}=sin\,x\) if \(n\) is a multiple of \(4\)

\(\dfrac {d^ny}{dx^n}=cos\,x\) if \(n\) is a multiple of \(4n+1\)

\(\dfrac {d^ny}{dx^n}=-sin\,x\) if \(n\) is a multiple of \(4n+2\)

\(\dfrac {d^ny}{dx^n}=-cos\,x\) if \(n\) is a multiple of \(4n+3\)

Illustration Questions

Find the \(35^{th}\) derivative of \(y=sin\,x\)

A \(sin\,x\)

B \(-cos\,x\)

C \(tan\,x\)

D \(sec^2\,x\)

×

\(35=4×8+3\rightarrow\) is of the form \(4n+3\)

\(\therefore \dfrac {d^{35}}{dx^{35}}\,(sin\,x)=-cos\,x\)

Find the \(35^{th}\) derivative of \(y=sin\,x\)

A

\(sin\,x\)

.

B

\(-cos\,x\)

C

\(tan\,x\)

D

\(sec^2\,x\)

Option B is Correct

Finding Higher Order Derivative of cos x by Observing a Pattern

Observe that:

\(\dfrac {d}{dx}(cos\,x)=-sin\,x\)

\(\Rightarrow\dfrac {d^2}{dx^2}(cos\,x)=-cos\,x\)

\(\Rightarrow\dfrac {d^3}{dx^3}\,(cos\,x)=sin\,x\)

  \(\Rightarrow\dfrac {d^4}{dx^4}\,(cos\,x)=cos\,x\)

If \(y=cos\,x\)

\(\therefore\)  \(\dfrac {d^ny}{dx^n}=cos\,x\) if \(n\) is a multiple of \(4\)

\(\dfrac {d^ny}{dx^n}=-sin\,x\) if \(n\) is a multiple of \(4n+1\)

\(\dfrac {d^ny}{dx^n}=-cos\,x\) if \(n\) is a multiple of \(4n+2\)

\(\dfrac {d^ny}{dx^n}=sin\,x\) if \(n\) is a multiple of \(4n+3\)

Illustration Questions

Find the \(81^{th}\) derivative of \(y=cos\,x\).

A \(-sin\,x\)

B \(cos\,x\)

C \(tan\,x\)

D \(sec\,x\)

×

\(81=4×20+1\rightarrow\) is of the form \(4n+1\)

\(\therefore \dfrac {d^{81}}{dx^{81}}\,(cos\,x)=-sin\,x\)

Find the \(81^{th}\) derivative of \(y=cos\,x\).

A

\(-sin\,x\)

.

B

\(cos\,x\)

C

\(tan\,x\)

D

\(sec\,x\)

Option A is Correct

Nth Derivative of sin(mx) 

\(\Rightarrow f(x) =sin\,m\,x\)

\(\Rightarrow f'(x) = m\,cos\,m\,x \)

\(\Rightarrow m\,sin \left(m\,x+\dfrac{\pi}{2}\right)\)

\(f''(x) = -m^2\,sin\,m\,x \Rightarrow m^2 \,sin(m\,x+\pi)\)

\(f''' (x) = -m^3\,cos\,m\,x\Rightarrow m^3\,sin \left(\dfrac{3\pi}{2}+m\,x\right)\)

Nth derivative \( sin\,m\,x = m^n\,sin \left(m\,x +\dfrac{n\,x}{2}\right)\) 

Illustration Questions

Find 7 th   derivative of  \(sin\,2\,x\) .

A \(2^6\,cos\,2x\)

B \(-2^7 \,cos\,2x \)

C \(cos\,2x\)

D \(0\)

×

Here m=2 ,       n= 7 

\(\Rightarrow\dfrac{d^7y}{d\,x^7} = 2^7\,sin\,\left(2x+\dfrac{7\,\pi}{2}\right)\)

\(= -2^7\,cos\,2\,x\)

Find 7 th   derivative of  \(sin\,2\,x\) .

A

\(2^6\,cos\,2x\)

.

B

\(-2^7 \,cos\,2x \)

C

\(cos\,2x\)

D

\(0\)

Option B is Correct

Nth Derivative of cos(mx) 

\(\because f(x) =cos\,m\,x\)

\(\therefore f'(x) = -m\,sin\,m\,x\)

\(= m\,cos\left(m\,x-\dfrac{\pi}{2}\right)\)

\(f''(x) = -m^2\,cos (m\,x)\)

\(= m^2cos(m\,x-\pi)\)

\(f''' (x) = m^3\,sin(m\,x)\)

\(= m^3 \,cos\left(m\,x-\dfrac{3\,\pi}{2}\right)\)

\(N^{th}\) derivative of  \(cos\,m\,x \)       (so \(\forall\,n\in I \))

\(=m^n\,cos\left(m\,x-\dfrac{n\,\pi}{2}\right)\)

Illustration Questions

Find 7th derivative of \(cos\,3\,x\) .

A \(0\)

B \(3^7\,cos\,3\,x\)

C \(3^7\,sin\,\,x\)

D \(3^7\,sin\,3\,x\)

×

Here m= 3,    n= 7

\(\Rightarrow\dfrac{d^7\,y}{d\,x^7} = 3^7 \,cos\left(3x-\dfrac{7\,\pi}{2}\right)\)

\(= 3^7\,sin\,3\,x\)

Find 7th derivative of \(cos\,3\,x\) .

A

\(0\)

.

B

\(3^7\,cos\,3\,x\)

C

\(3^7\,sin\,\,x\)

D

\(3^7\,sin\,3\,x\)

Option D is Correct

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