Understanding Engineering Mathematics

B. What are the most general functions that you need to differentiate to obtain the
following functions?

(i) x^4 (ii) cosx (iii) ex (iv) sinx

(v)

1

x^4

(vi)

√

x (vii)

1

x

(viii) 0

(ix)

1

cos^2 x

8.3.4 Rules of differentiation

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A.Using the definition of the functions and appropriate rules of differentiation obtain
the derivatives of the following elementary functions (see Section 4.4 for hyperbolic
functions).

(i) secx (ii) cosecx (iii) cotx (iv) coshx

(v) sinhx (vi) tanhx (vii) cosechx (viii) sechx

(ix) cothx

B. Differentiate

(i) ln(secx) (ii) ln(sinx) (iii) ln(secx+tanx)

(iv) ln(cosecx+cotx) (v) ln(coshx)

(vi) ln(sinhx)

C.Differentiate

(i) x^7 − 2 x^5 +x^4 −x^2 + 2 (ii) (x^2 + 2 )tanx

(iii)

lnx

x^2 + 1

(iv) exp(x^3 − 2 x) (v)

x

√

x^2 − 1

(vi) ln(cosx+ 1 ) (vii) sin

(

x+ 1

x

)

(viii) secxtanx

(ix) e^6 x (x) xex (xi) e−x

2

(xii) ln 5x (xiii) exlnx (xiv) lne^2 x

8.3.5 Implicit differentiation

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A.Use implicit differentiation to differentiate the functions

(i) cos−^1 x (ii) tan−^1 x (iii) πx

B. Evaluatedy/dxat the points indicated.

(i) x^2 +y^2 = 1 ( 0 , 1 ) (ii) x^3 − 2 x^2 y+y^2 = 1 ( 1 , 2 )

C.Iff(x)=

x+ 1

x− 3

,evaluatef′( 0 ).