Standard Derivatives

Standard Derivatives

(i) Algebraic Functions:

a) \(\frac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}}\), \(\frac{d}{dx}{{\left\{ f\left( x \right) \right\}}^{n}}=n{{\left\{ f\left( x \right) \right\}}^{n-1}}\frac{d}{dx}f\left( x \right)\).

b) \(\frac{d}{dx}\left[ \frac{1}{{{x}^{n}}} \right]=-n{{x}^{-n-1}}\), \(\frac{d}{dx}{{\left\{ \frac{1}{f\left( x \right)} \right\}}^{n}}=\frac{-n}{{{\left\{ f\left( x \right) \right\}}^{n-1}}}\frac{d}{dx}f\left( x \right)\).

c) \(\frac{d}{dx}\left( \sqrt{x} \right)=\frac{1}{2\sqrt{x}}\), \(\frac{d}{dx}\sqrt{f\left( x \right)}=\frac{1}{2\sqrt{f\left( x \right)}}\frac{d}{dx}f\left( x \right)\).

(ii) Exponential Functions:

a) \(\frac{d}{dx}\left( {{e}^{x}} \right)={{e}^{x}}\), \(\frac{d}{dx}{{e}^{f\left( x \right)}}=\left[ {{e}^{f\left( x \right)}} \right]\frac{d}{dx}f\left( x \right)\).

b) \(\frac{d}{dx}\left( {{a}^{x}} \right)={{a}^{x}}{{\log }_{e}}a\), \(\frac{d}{dx}\left[ {{a}^{f\left( x \right)}} \right]=\left[ {{a}^{f\left( x \right)}}{{\log }_{e}}a \right]\frac{d}{dx}f\left( x \right)\).

(iii) Logarithmic Functions:

a) \(\frac{d}{dx}\left( {{\log }_{e}}x \right)=\frac{1}{x}\), \(\frac{d}{dx}\left[ {{\log }_{e}}f\left( x \right) \right]=\frac{1}{f\left( x \right)}\frac{d}{dx}f\left( x \right)\).

b) \(\frac{d}{dx}\left( {{\log }_{a}}x \right)=\frac{1}{x{{\log }_{e}}a}\), \(\,\frac{d}{dx}\left[ {{\log }_{a}}f\left( x \right) \right]=\frac{1}{f\left( x \right){{\log }_{e}}a}\frac{d}{dx}f\left( x \right)\).

(iv) Trigonometric Functions:

a) d/dx (sin x) = cosx

b) d/dx (cos x) = – sin x

c) d/dx (tan x) = sec²x

d) d/dx (cot x) = – cosec²x

e) d/dx (sec x) = sec x. tan x

f) d/dx (cosec x) = – cosec x. cot x.

(v) Inverse Trigonometric Functions:

a) \(\frac{d}{dx}\left( {{\sin }^{-1}}x \right)=\frac{1}{\sqrt{1-{{x}^{2}}}}\), -1 < x < 1.

b) \(\frac{d}{dx}\left( {{\cos }^{-1}}x \right)=\frac{-1}{\sqrt{1-{{x}^{2}}}}\), -1 < x < 1.

c) \(\frac{d}{dx}\left( {{\tan }^{-1}}x \right)=\frac{1}{1+{{x}^{2}}}\), – ∞ < x < ∞.

d) \(\frac{d}{dx}\left( {{\cot }^{-1}}x \right)=\frac{-1}{1+{{x}^{2}}}\).

e) \(\frac{d}{dx}\left( {{\sec }^{-1}}x \right)=\frac{1}{\left| x \right|\sqrt{{{x}^{2}}-1}}\), |x| > 1.

f) \(\frac{d}{dx}\left( co{{\sec }^{-1}}x \right)=\frac{-1}{\left| x \right|\sqrt{{{x}^{2}}-1}}\).

Logarithmic Differentiation: In this section, we will be mainly discussing derivatives of the function of the form [f(x)]g(x) where f(x) and g(x) are functions of x To find the derivative of this type of function we proceed as follows:

Let y = [f(x)]g(x)

Taking logarithm of both the sides, we have

logy = g(x). log [f(x)].

Differentiating w.r.t x we get

\(\frac{1}{y}\frac{dy}{dx}=g\left( x \right).\frac{1}{f\left( x \right)}\frac{df\left( x \right)}{dx}+\log f\left( x \right).\frac{dg\left( x \right)}{dx}\).

∴ \(\frac{dy}{dx}=y\left[ \frac{g\left( x \right)}{f\left( x \right)}.\frac{df\left( x \right)}{dx}+\log \left[ f\left( x \right) \right].\frac{dg\left( x \right)}{dx} \right]\).

Alternative: We have,

y = [f (x)]g(x)

= e g(x) log [f(x)]

\(\frac{dy}{dx}={{e}^{g\left( x \right)\log \left[ f\left( x \right) \right]}}\left[ g\left( x \right).\frac{1}{f\left( x \right)}\frac{df\left( x \right)}{dx}+\log \left[ f\left( x \right) \right]\frac{dg\left( x \right)}{dx} \right]\).

= \({{\left[ f\left( x \right) \right]}^{g\left( x \right)}}\left[ \frac{g\left( x \right)}{f\left( x \right)}.\frac{df\left( x \right)}{dx}+\log \left[ f\left( x \right) \right].\frac{dg\left( x \right)}{dx} \right]\).

Example: Find the derivative of x

Solution: Let y = xx

logy = xlogx

Differentiate w.r.t x

1/y dy/dx = logx + x.1/x

dy/dx = y [logx + 1]

∴ dy/dx = xx [log x + 1].