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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

Syllabus

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

EASY

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

$\sqrt{2}$

$\frac{1}{\sqrt{2}}$

$\frac{1}{2}$

$1$

View Solution

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)

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

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

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