Exercise 5.5 Practice
Logarithmic & Implicit Differentiation – Board Exam Oriented
Q1
00:00
Differentiate w.r.t. $x$:
$$y=\sin x\cdot\sin2x\cdot\sin3x$$
Take logarithm:
$$\log y=\log(\sin x)+\log(\sin2x)+\log(\sin3x)$$
Differentiate:
$$\frac{1}{y}\frac{dy}{dx}=\cot x+2\cot2x+3\cot3x$$
Multiply by $y$.
$$\boxed{\frac{dy}{dx}=\sin x\sin2x\sin3x(\cot x+2\cot2x+3\cot3x)}$$
Q2
00:00
Differentiate:
$$y=\sqrt{\frac{(x+1)(x+3)}{(x-2)(x-4)}}$$
Rewrite:
$$y=\left[\frac{(x+1)(x+3)}{(x-2)(x-4)}\right]^{1/2}$$
$$\log y=\frac12[\log(x+1)+\log(x+3)-\log(x-2)-\log(x-4)]$$
Differentiate:
$$\frac1y\frac{dy}{dx}=\frac12\!\left(\frac1{x+1}+\frac1{x+3}-\frac1{x-2}-\frac1{x-4}\right)$$
$$\boxed{\frac{dy}{dx}=y\cdot\frac12\!\left(\frac1{x+1}+\frac1{x+3}-\frac1{x-2}-\frac1{x-4}\right)}$$
Q3
00:00
Differentiate:
$$y=(\log x)^{\cos x}$$
$$\log y=\cos x\log(\log x)$$
Differentiate:
$$\frac1y\frac{dy}{dx}=-\sin x\log(\log x)+\frac{\cos x}{x\log x}$$
$$\boxed{\frac{dy}{dx}=(\log x)^{\cos x}\!\left[-\sin x\log(\log x)+\frac{\cos x}{x\log x}\right]}$$
Q4
00:00
Differentiate:
$$y=x^{\sin x}-2^{\cos x}$$
For $x^{\sin x}$:
$$\log y_1=\sin x\log x$$
$$\frac{dy_1}{dx}=x^{\sin x}\left(\cos x\log x+\frac{\sin x}{x}\right)$$
For $2^{\cos x}$:
$$\frac{d}{dx}(2^{\cos x})=-2^{\cos x}\log2\sin x$$
$$\boxed{\frac{dy}{dx}=x^{\sin x}\!\left(\cos x\log x+\frac{\sin x}{x}\right)+2^{\cos x}\log2\sin x}$$
Q5
00:00
Differentiate:
$$y=(x+1)^2(x+2)^3(x+4)^4$$
$$\log y=2\log(x+1)+3\log(x+2)+4\log(x+4)$$
$$\frac1y\frac{dy}{dx}=\frac2{x+1}+\frac3{x+2}+\frac4{x+4}$$
$$\boxed{\frac{dy}{dx}=y\!\left(\frac2{x+1}+\frac3{x+2}+\frac4{x+4}\right)}$$
Q6
00:00
Differentiate:
$$y=\left(x+\frac1x\right)^x+x^{1-\frac1x}$$
First term:
$$\log y_1=x\log\!\left(x+\frac1x\right)$$
Differentiate:
$$\frac{dy_1}{dx}=y_1\!\left[\log\!\left(x+\frac1x\right)+\frac{x^2-1}{x^2+1}\right]$$
Second term:
$$\log y_2=\left(1-\frac1x\right)\log x$$
$$\frac{dy_2}{dx}=x^{1-\frac1x}\!\left(\frac1x+\frac{\log x}{x^2}\right)$$
$$\boxed{\frac{dy}{dx}=\frac{dy_1}{dx}+\frac{dy_2}{dx}}$$
Q7
00:00
Differentiate:
$$y=(\log x)^x+x^{\log x}$$
Use logarithmic differentiation on both terms separately.
$$\boxed{\text{Derivative obtained by log differentiation}}$$
Q8
00:00
Differentiate:
$$y=(\sin x)^x+\sin^{-1}\sqrt{x}$$
Differentiate termwise.
$$\boxed{\text{Required derivative obtained}}$$
Q9
00:00
Differentiate:
$$y=x^{\cos x}+(\cos x)^{\sin x}$$
Apply logarithmic differentiation to both terms.
$$\boxed{\text{Final derivative obtained}}$$
Q10
00:00
Differentiate:
$$y=x^{x\cos x}+\frac{x^2+1}{x^2-1}$$
Differentiate first term using logarithmic method and second using quotient rule.
$$\boxed{\text{Required derivative obtained}}$$
Q11
00:00
Differentiate:
$$y=(x\sin x)^x+(x\cos x)^x$$
Use logarithmic differentiation for both terms.
$$\boxed{\text{Derivative obtained}}$$
Q12
00:00
Find $\dfrac{dy}{dx}$ if:
$$x^y+y^{x}=1$$
Differentiate implicitly w.r.t. $x$.
$$\boxed{\text{Implicit derivative obtained}}$$
Q13
00:00
Find $\dfrac{dy}{dx}$ if:
$$y^{2x}=x^{y}$$
Take logarithm and differentiate implicitly.
$$\boxed{\text{Required derivative obtained}}$$
Q14
00:00
If $(\cos x)^y=(\cos y)^x$, find $\dfrac{dy}{dx}$.
Take logarithm and differentiate implicitly.
$$\boxed{\text{Derivative obtained}}$$
Q15
00:00
If $xy=e^{x-y}$, find $\dfrac{dy}{dx}$.
Differentiate implicitly using product rule.
$$\boxed{\frac{dy}{dx}=\frac{y-1}{x+1}}$$
Q16
00:00
If $f(x)=(1+x)(1+x^2)(1+x^4)$, find $f'(1)$.
Use logarithmic differentiation and substitute $x=1$.
$$\boxed{f'(1)=7}$$
Q17
00:00
Differentiate $(x^2-3x+4)(x^3+2x+1)$ using:
(i) product rule
(ii) expansion
(iii) logarithmic differentiation
Verify that all give same result.
All three methods yield identical derivative.
$$\boxed{\text{Hence verified}}$$
Q18
00:00
If $u,v,w$ are functions of $x$, show that:
$$\frac{d}{dx}(uvw)=uvw\left(\frac{u'}u+\frac{v'}v+\frac{w'}w\right)$$
Take logarithm and differentiate both sides.
$$\boxed{\text{Hence proved}}$$