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Derivative of Polynomials and Trigonometric Functions - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Derivative of the Trigonometric Function (csc/sec/cot) is considered one of the most asked concept.

  • 121 Questions around this concept.

Solve by difficulty

QuestionEASY

If   y= \sec \left ( \tan ^{-1}x \right ),then\: \: \frac{dy}{dx}\: \: at\: x= 1 is equal to:

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Concepts Covered - 4

Derivative of the Polynomial Function

Derivative of the Polynomial Function

Finding derivatives of functions by using the definition of the derivative (by using First Principle) can be a lengthy and, for certain functions, a rather challenging process. In this section, we will learn direct results for finding derivatives of certain standard functions that allow us to bypass this process.

\\\mathbf{1.}\;\;\;\;\mathrm{\mathbf{\frac{\mathit{d}}{\mathit{dx}}(constant)=0}}
\\f(x)=c\;\;and\;\;f(x+h)=c \\\\\begin{aligned}f^{\prime}(x) &=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\ &=\lim _{h \rightarrow 0} \frac{c-c}{h} \\ &=\lim _{h \rightarrow 0} \frac{0}{h} \\ &=\lim _{h \rightarrow 0} 0=0 \end{aligned}

So the derivative of a constant function is zero. Also since a constant function is a horizontal line the slope of a constant function is 0.

\\\mathbf{2.}\;\;\;\;\mathrm{\mathbf{\frac{\mathit{d}}{\mathit{dx}}(x^n)=nx^{n-1}}}

For f(x) = xn where n is a positive integer, we have

\\f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{(x+h)^{n}-x^{n}}{h}

Since,

(x+h)^{n}=x^{n}+n x^{n-1} h+\left(\begin{array}{l}{n} \\ {2}\end{array}\right) x^{n-2} h^{2}+\left(\begin{array}{l}{n} \\ {3}\end{array}\right) x^{n-3} h^{3}+\ldots+n x h^{n-1}+h^{n}

we can write

(x+h)^{n}-x^{n}=n x^{n-1} h+\left(\begin{array}{l}{n} \\ {2}\end{array}\right) x^{n-2} h^{2}+\left(\begin{array}{l}{n} \\ {3}\end{array}\right) x^{n-3} h^{3}+\ldots+n x h^{n-1}+h^{n}

Next, divide both sides by h:

\frac{(x+h)^{n}-x^{n}}{h}=\frac{n x^{n-1} h+\left(\begin{array}{l}{n} \\ {2}\end{array}\right) x^{n-2} h^{2}+\left(\begin{array}{l}{n} \\ {3}\end{array}\right) x^{n-3} h^{3}+\ldots+n x h^{n-1}+h^{n}}{h}

Thus,

\frac{(x+h)^{n}-x^{n}}{h}=n x^{n-1}+\left(_{2}^{n}\right) x^{n-2} h+\left(_{3}^{n}\right) x^{n-3} h^{2}+\ldots+n x h^{n-2}+h^{n-1}

Finally,

\\f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{(x+h)^{n}-x^{n}}{h}\\\\\begin{aligned} \;\;\;\;\;\;\; &=\lim _{h \rightarrow 0}\left(n x^{n-1}+\left(_{2}^{n}\right) x^{n-2} h+\left(_{3}^{n}\right) x^{n-3} h^{2}+\ldots+n x h^{n-1}+h^{n}\right) \\\\ &=n x^{n-1} \end{aligned}

Derivative of the Logarithm and Exponential Function

Derivative of Exponential and Logarithmic Functions

\\\mathbf{4.}\;\;\;\;\mathrm{\mathbf{\frac{\mathit{d}}{\mathit{dx}}(a^x)=a^x\log_ea}}

\\f(x)=a^x\;\;and\;\;f(x+h)=a^{x+h} \\\\\begin{aligned}f^{\prime}(x) &=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\ &=\lim _{h \rightarrow 0} \frac{a^{x+h}-a^x}{h} \\ &=a^x\lim _{h \rightarrow 0} \frac{a^h-1}{h} \\ &=a^x\log_ea\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left [ \because \lim _{x \rightarrow 0} \frac{a^x-1}{x}=\log_ea \right ] \end{aligned}

\\\mathbf{5.}\;\;\;\;\mathrm{\mathbf{\frac{\mathit{d}}{\mathit{dx}}(e^x)=e^x\log_ee=e^x}}

 

\\\mathbf{6.}\;\;\;\;\mathrm{\mathbf{\frac{\mathit{d}}{\mathit{dx}}(\log_e|x|)=\frac{1}{x},\;\;x\neq0}}

\\f(x)=\log_e(x)\;\;and\;\;f(x+h)=\log_e(x+h) \\\\\begin{aligned}f^{\prime}(x) &=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\ &=\lim _{h \rightarrow 0} \frac{\log_e(x+h)-\log_e(x)}{h} \\ &=\lim _{h \rightarrow 0} \frac{\log_e\left ( \frac{x+h}{x} \right )}{h}\\&=\lim _{h \rightarrow 0} \frac{\log_e\left (1+ \frac{h}{x} \right )}{\frac{h}{x}\cdot x} \\ &=\frac{1}{x}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left [ \because \lim _{x \rightarrow 0} \frac{\log_e (1+x)}{x}=1 \right ] \end{aligned}

\\\mathbf{7.}\;\;\;\;\mathrm{\mathbf{\frac{\mathit{d}}{\mathit{dx}}(\log_a|x|)=\frac{1}{x\log_e a},\;\;x\neq0}}

Derivative of the Trigonometric Function (cos/sine/tan)

Derivative of the Trigonometric Functions (sin/cos/tan)

\\\mathbf{8.}\;\;\;\;\mathrm{\mathbf{\frac{\mathit{d}}{\mathit{dx}}(\sin(x))=\cos(x)}}

\\f(x)=\sin(x)\;\;and\;\;f(x+h)=\sin(x+h) \\\\\begin{aligned}f^{\prime}(x) &=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\ &=\lim _{h \rightarrow 0} \frac{\sin(x+h)-\sin(x)}{h} \\ &=\lim _{h \rightarrow 0} \frac{2\sin\frac{h}{2}\cdot\cos\left ( \frac{2x+h}{2} \right )}{h}\\&=\lim _{h \rightarrow 0} \frac{\sin\frac{h}{2}}{\frac{h}{2}}\cdot\lim_{h\rightarrow 0}{\cos\left ( \frac{2x+h}{2} \right )} \\ &=\cos x\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left [ \because \lim _{x \rightarrow 0} \frac{\sin (x)}{x}=1 \right ] \end{aligned}

\\\mathbf{9.}\;\;\;\;\mathrm{\mathbf{\frac{\mathit{d}}{\mathit{dx}}(\cos(x))=-\sin(x)}}

\\\mathbf{10.}\;\;\;\;\mathrm{\mathbf{\frac{\mathit{d}}{\mathit{dx}}(\tan(x))=\sec^2(x)}}

Derivative of the Trigonometric Function (csc/sec/cot)

Derivative of the Trigonometric Functions (cot/sec/csc)

\\\mathbf{11.}\;\;\;\;\mathrm{\mathbf{\frac{\mathit{d}}{\mathit{dx}}(\cot(x))=-\csc^2(x)}}

\\\mathbf{12.}\;\;\;\;\mathrm{\mathbf{\frac{\mathit{d}}{\mathit{dx}}(\sec(x))=\sec (x)\tan (x)}}

\\\mathbf{13.}\;\;\;\;\mathrm{\mathbf{\frac{\mathit{d}}{\mathit{dx}}(\csc(x))=-\csc (x)\cot (x)}}

Study it with Videos

Derivative of the Polynomial Function
Derivative of the Logarithm and Exponential Function
Derivative of the Trigonometric Function (cos/sine/tan)

Study other Related Concepts

Derivative of Polynomials and Trigonometric Functions Current Topic

Limit
Left-Hand Limits and Right-Hand Limits
Algebra of Limits
Limit of Indeterminate Form and Algebraic limit
Limit of Algebraic function
Limit Using Expansion
Sandwich Theorem
Limits of Trigonometric Functions
Exponential and Logarithmic Limits
Limits of the form (1 power infinity)

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Books

Reference Books

Derivative of the Polynomial Function

Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

Page No. : 3.2

Line : 41

Derivative of the Logarithm and Exponential Function

Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

Page No. : 3.3

Line : 1

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