Learn derivatives of logarithmic functions and logarithmic differentiation examples, finding the derivative of expressions and equation of tangent to a curve involving log function at particular values of x.

- If \(y=\ell n\,x\), then \(\dfrac {dy}{dx}=\dfrac {1}{x}\)

\(\Rightarrow \dfrac {d}{dx}\,(\ell n\,x)=\dfrac {1}{x}\)

- The derivative of \(\ell n\,x\) w.r.t. \(x\) is \(\dfrac {1}{x}\).
- If \(u(x)\) is a function of x, then \(\dfrac {d}{dx}\,(\ell n\,u)=\dfrac {1}{u}\,\dfrac {du}{dx}\) (by Chain Rule)

e.g. \(\dfrac {d}{dx}\Big(\ell n\,(cot\,x)\Big)=\dfrac {1}{cot\,x}×\dfrac {d}{dx}\,cot\,x\)

\(=\dfrac {1}{cot\,x}×(-cosec^2x)=\dfrac {-1}{sin\,x\,cosx}\)

- The above can also be written as :\(\dfrac {d}{dx}\left (\ell n (g(x)\right)=\dfrac {1}{g(x)}×g'(x)\)

A \(\dfrac {8x+5}{2x^2+3x+1}\)

B \(\dfrac {2x+7}{x^2+x+1}\)

C \(\ell n\,x+e^x\)

D \(\dfrac {4x+3}{2x^2+3x+1}\)

- \(f'(a)\) means the value of derivative of the function \('f'\) at \(x=a\).

A \(\dfrac {1}{12}\)

B \(\dfrac {1}{4}\)

C \(\dfrac {5}{6}\)

D \(\dfrac {7}{12}\)

- Sometimes we face complicated functions which involve products, quotients or powers of some terms, these can often be simplified by taking logarithm and then differentiated. This is called logarithmic differentiation.
- e.g.

If \(y=\dfrac {(x+1)^3×\sqrt [4] {x-2}}{\sqrt [5] {(x-3)^2}} \) and we have to find \(\dfrac{dy}{dx}\).

We take log both sides to base e.

\(\therefore\ell n\;y=\ell n\left(\dfrac {(x+1)^3×\sqrt [4] {x-2}}{\sqrt [5] {(x-3)^2}}\right) \) (Now, use properties of log)

\(=3\,\ell n (x+1)+\dfrac {1}{4}\ell n(x-2)-\dfrac {2}{5}\ell n\, (x-3)\)

Now, observe that we can differentiate both sides easily.

\(\Rightarrow \dfrac {1}{y}\,\dfrac {dy}{dx}=\dfrac {3}{x+1}+\dfrac {1}{4(x-2)}-\dfrac {2}{5(x-3)}\)

\(\Rightarrow \dfrac {dy}{dx}=y\left [\dfrac {3}{x+1}+\dfrac {1}{4(x-2)}-\dfrac {2}{5(x-3)}\right]\)

\(=\dfrac {(x+1)^3×\sqrt [4] {x-2}}{\sqrt [5] {(x-3)^2}} \left [\dfrac {3}{x+1}+\dfrac {1}{4(x-2)}-\dfrac {2}{5(x-3)}\right]\)

A \(\dfrac {(x-2)^2×(x+1)^{1/3}}{(x-5)^3} \left [\dfrac {2}{x-2}+\dfrac {1}{3(x+1)}-\dfrac {3}{x-5}\right]\)

B \(3\,sin\,x-2cos^2\,x+\dfrac {1}{x}\)

C \(\dfrac {(x-2)^2×\sqrt [3] {x+1}}{ {(x-5)^3}} \left [\dfrac {3}{x-2}+\dfrac {1}{x-5}\right]\)

D \(\dfrac {3}{sin\,x}-\dfrac {2}{x+1}+cos^3\,x\)

- Sometimes we come across functions of the form \(y=(f(x))^{g(x)}\), to differentiate function of these form we first take log on both sides and then differentiate.
- \(y=(f(x))^{g(x)}\Rightarrow\) Take log on both sides to base e

\(\Rightarrow \ell n\,y=\ell n(f(x))^{g(x)}\)

\(\Rightarrow \ell n\,y=g(x)\;\ell n(f(x))\)

\(\Rightarrow \dfrac {1}{y}\,\dfrac {dy}{dx}=\underbrace{g'(x)\,\ell n\,f(x)+\dfrac {g(x)}{f(x)}f'(x)}_{Product\, rule}\)

\(\Rightarrow \dfrac {dy}{dx}=(f(x))^{g(x)}\Big[g'(x)\,\ell n\,f(x)+\dfrac {g(x)}{f(x)}f'(x)\Big]\)?

- Do not remember the formula but do these steps for the \('f'\) and \('g'\) given in the problem.

A \((cos\,x)^{x^2} [-x^2\,tan\,x+2x\,ln\,(cos\,x)]\)

B \((cos\,x)^{x^2} [-x\,tan\,x+x\,\ell n(cosx)]\)

C \((sin\,x)^{cos\,x}\,(1+2x)\)

D \((tan\,x)\,(e^x+\ell n\,x)\)

- Equation of tangent to \(y=f(x)\) at a point \((a, f(a))\) on it is \(y-f(a)=f'(a)(x-a)\)

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

B \(5x-3y-7=0\)

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

D \(9x+y-9=0\)

*\(\dfrac {d}{dx}(log_a\,x)=\dfrac {d}{dx}\underbrace {\left (\dfrac {\ell n\, x}{\ell n\,a}\right)}_{\text {Base change formula}} =\dfrac {1}{\ell n\,a}×\dfrac {d}{dx}(\ell n\,x)=\dfrac {1}{x\,\ell n\,a}\)*

\(\therefore \;\dfrac {d}{dx}(log_a\,x)=\dfrac {1}{x\,\ell n\,a}\)

- If we take \(a=e\), then it reduces to \(\dfrac {d}{dx}(\ell n\,x)=\dfrac {1}{x}\)

\( \;\dfrac {d}{dx}(log_a\,f(x))=\underbrace{\dfrac {1}{\ell n\,a}× \dfrac {1}{f(x)}×f'(x)=\dfrac {f'(x)}{f(x)\,\ell n\,a}}_{chain\, rule} \)

\(\therefore \;\dfrac {d}{dx}(log_a\,f(x)) ={\dfrac {1}{f(x)\,\ell n\,a}× f'(x)}\)

A \(\ell n\,x-x^2+x^3+C\)

B \(\dfrac {(cos\,x+2sin\,x)}{(\ell n\,2)(sin\,x)}\)

C \(\dfrac {(2\,cos\,x-sin\,x)}{(\ell n\,2)(cos\,x)}\)

D \(e^{log^{(sin\,x)}}+cos\,x+C\)

**\(f''(x)=\dfrac {d^2y}{dx^2}=\dfrac {d}{dx}\left (\dfrac {dy}{dx}\right)\)**- In general, \(f^n(x)=\dfrac {d^ny}{dx^n}=n^{th}\) derivative of \(f\) with respect to \(x\).
- Treat \(f'\) as a function , differentiate it again to get \(f''(x)\) and so on.

A \(x(5+6\,\ell n\,x)\)

B \(x^2(6+5\,\ell n\,x)\)

C \((sin\,x)(2+x^2)\)

D \(\dfrac {\ell n\,x}{x}\)

- Suppose \(f(x)=Cx^2-\ell n\,x\)

If \(f'(1)=1\), then

\(\Rightarrow2C\,x-\dfrac {1}{x}=1\)

\(\Rightarrow2C-1=1\)

\(\Rightarrow C=1\)

So, \(C=1\), when \(f'(1)=1\)

A \(\alpha =2\)

B \(\alpha =-2\)

C \(\alpha =5\)

D \(\alpha =1\)