Understanding Engineering Mathematics

=lim δx→ 0 +δx = 8.1.3 Standard derivatives ➤232 243➤➤ Give the derivatives of the following functions (i) 49 (ii) x^4 (iii) √ x ...

B.Ifx=t^2 +1, y=t−1, evaluate d^2 y dx^2 as a function oft. 8.2 Revision 8.2.1 Geometrical interpretation of differentiation ➤ 2 ...

y = f(x) y + dy = f(x + dx) dy = f(x + dx) − f(x) x dx x y y = f(x) A B Figure 8.2 The gradient (210 ➤ ) of the extended chordAB ...

corresponding increase iny: δy=f(x+δx)−f(x) and neglect any powers ofδxgreater than one to get something like δyf′(x)δx wheref′ ...

Ta b l e 8. 1 Functionf(x) Derivativef′(x) c=constant 0 xn nxn−^1 n = 0 xn+^1 /(n+ 1 )xn(n =− 1 ) ln|x|=loge|x| 1 /x sinx cosx ...

This is, of course, an incorrect application of the rule for the deriva- tive ofxn. We will come back to 2xlater (Section 8.2.5) ...

uandvwill increase tou+δuandv+δurespectively, withδuandδvvery small. The area therefore increases to (u+δu)(v+δv)=uv+vδu+uδv+δuδ ...

convenient to do this – with care. For example in differential equations we sometimes use such steps as: dy dx dx≡dy which is re ...

So by product rule, with sayu=x^2 ,v=cosx, dy dx = d(uv) dx = d dx (x^2 cosx) = vdu dx + udv dx = d(x^2 ) dx cosx+x^2 d dx (cosx ...

(vi)y=e−^2 x This is a function of a function. Here we will practise a short hand approach to the function of a function rule. d ...

so 2 x+ 2 y+ 2 x dy dx + 4 y dy dx = 0 Hence(x+y)+(x+ 2 y) dy dx = 0 and so dy dx =− x+y x+ 2 y B.This is a standard application ...

You should now be able to repeat this argument with the special case ofa=2 to obtain d dx ( 2 x)= 2 xln 2 D.Ify=f(x)= x− 1 x+ 2 ...

Solution to review question 8.1.6 Withx= 3 t^2 ,y=cos(t+ 1 )we have dy dt =−sin(t+ 1 )and dx dt = 6 t,so: dy dx = dy/dt dx/dt = ...

(iii) d^2 dx^2 (e−xcosx)= d dx (−e−xcosx−e−xsinx) =− d dx (e−x(cosx+sinx)) =−[−e−x(cosx+sinx)+e−x(−sinx+cosx)] =−e−x(−2sinx) = 2 ...

function of a function rule to change the differentiation to one with respect tot, and write d^2 y dx^2 = d dx ( dy dx ) = d dt ...

B. What are the most general functions that you need to differentiate to obtain the following functions? (i) x^4 (ii) cosx (iii) ...

8.3.6 Parametric differentiation ➤➤ 229 240 ➤ A.Ifx=e^2 t,y=et+1, evaluate dy dx and d^2 y dx^2 as functions oftby two different ...

(iii) Discuss the forms of motion described by the relations between position,s,and time,t, given below. (a)s=at+b (b)s=at^2 +bt ...

4.In a circuit with an inductor of inductanceLand capacitor of capacitanceC, the voltage across the inductance is given by V=L d ...

8.3.2 Differentiation from first principles (i) 3 (ii) 2x+ 2 (iii) 3x^2 (iv) −sinx 8.3.3 Standard derivatives A.(i) ex (ii) −sin ...