Bird's Higher Engineering Mathematics

John Bird

Chapter 25 Methods of differentiation - all with Video Answers

Educators

Chapter Questions

02:04

Problem 1

Differentiate from first principle $f(x)=x^{2}$ and determine the value of the gradient of the curve at $x=2$

William Mead

Numerade Educator

02:59

Problem 1

If $y=3 x^{4}+2 x^{3}-3 x+2$ find
(a) $\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$
(b) $\frac{\mathrm{d}^{3} y}{\mathrm{~d} x^{3}}$

Mirza Aslam Beig

Numerade Educator

01:10

Problem 2

(a) Given $f(t)=\frac{2}{5} t^{2}-\frac{1}{t^{3}}+\frac{3}{t}-\sqrt{t}+1$ determine $f^{\prime \prime}(t)$
(b) Evaluate $f^{\prime \prime}(t)$ when $t=1$

Tyler Moulton

Numerade Educator

00:50

Problem 2

Find the differential coefficients of (a) $y=12 x^{3}$ (b) $y=\frac{12}{x^{3}}$

Varsha Aggarwal

Numerade Educator

01:01

The charge $q$ on the plates of a capacitor is given by $q=C V \mathrm{e}^{-\frac{t}{C R}}$, where $t$ is the time, $C$ is the capacitance and $R$ the resistance. Determine (a) the rate of change of charge, which is given by $\frac{\mathrm{d} q}{\mathrm{~d} t}$,
(b) the rate of change of current, which is given by $\frac{\mathrm{d}^{2} q}{\mathrm{~d} t^{2}}$

Narayan Hari

Numerade Educator

01:01

Problem 3

Differentiate (a) $y=6$ (b) $y=6 x$

Varsha Aggarwal

Numerade Educator

00:40

Problem 4

In Problems 4 and 5, find the second differential coefficient with respect to the variable.
$3 \sin 2 t+\cos t$ (b) $2 \ln 4 \theta$

Ernest Castorena

Numerade Educator

01:18

Problem 4

Find the derivatives of
(a) $y=3 \sqrt{x}$
(b) $y=\frac{5}{\sqrt[3]{x^{4}}}$

Varsha Aggarwal

Numerade Educator

01:36

Problem 5

In Problems 4 and 5, find the second differential coefficient with respect to the variable.
$2 \cos ^{2} x$ (b) $(2 x-3)^{4}$

Varsha Aggarwal

Numerade Educator

01:45

Problem 5

Differentiate, with respect to $x, y=5 x^{4}+4 x-\frac{1}{2 x^{2}}+\frac{1}{\sqrt{x}}-3$

Varsha Aggarwal

Numerade Educator

02:09

Problem 6

Evaluate $f^{\prime \prime}(\theta)$ when $\theta=0$ given $f(\theta)=2 \sec 3 \theta$

Amrita Bhasin

Numerade Educator

01:36

Problem 6

Find the differential coefficients of (a) $y=3 \sin 4 x$ (b) $f(t)=2 \cos 3 t$ with respect to the variable.

Varsha Aggarwal

Numerade Educator

02:54

Problem 7

Show that the differential equation $\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-4 \frac{\mathrm{d} y}{\mathrm{~d} x}+4 y=0$ is satisfied when $y=x \mathrm{e}^{2 x}$

Prashansha Kaushik

Numerade Educator

01:18

Problem 7

Determine the derivatives of (a) $y=3 \mathrm{e}^{5 x}$ (b) $f(\theta)=\frac{2}{\mathrm{e}^{3 \theta}}$ (c) $y=6 \ln 2 x$

Varsha Aggarwal

Numerade Educator

02:50

Problem 8

Show that, if $P$ and $Q$ are constants and $y=P \cos (\ln t)+Q \sin (\ln t)$, then
$$
t^{2} \frac{\mathrm{d}^{2} y}{\mathrm{~d} t^{2}}+t \frac{\mathrm{d} y}{\mathrm{~d} t}+y=0
$$

Gregory Higby

Numerade Educator

01:22

Problem 8

Find the gradient of the curve $y=3 x^{4}-2 x^{2}+5 x-2$ at the points $(0,-2)$ and $(1,4)$

Varsha Aggarwal

Numerade Educator

00:57

Problem 9

Determine the co-ordinates of the point on the graph $y=3 x^{2}-7 x+2$ where the gradient is $-1$

Varsha Aggarwal

Numerade Educator

01:38

Problem 9

The displacement, $s$, of a mass in a vibrating system is given by: $s=(1+t) \mathrm{e}^{-\omega t}$ where $\omega$ is the natural frequency of vibration. Show that: $\frac{\mathrm{d}^{2} s}{\mathrm{~d} t^{2}}+2 \omega \frac{\mathrm{d} s}{\mathrm{~d} t}+\omega^{2} s=0$

Given $y=2 x \mathrm{e}^{-3 x}$ show that
$$
\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+6 \frac{\mathrm{d} y}{\mathrm{~d} x}+9 y=0
$$

Varsha Aggarwal

Numerade Educator

01:47

Problem 27

Evaluate $\frac{\mathrm{d}^{2} y}{\mathrm{~d} \theta^{2}}$ when $\theta=0$ given $y=4 \sec 2 \theta$

Varsha Aggarwal

Numerade Educator