Find the derivative of:
(a) f(x)=ex(x2+1)3 (b) g(x)=(3x+24x2−5)5
Solution (a) The function f is the product of ex and (x2+1)3, so we first use the Product Rule. f′(x)=↑Product Ruleex[ddx(x2+1)3]+[ddxex](x2+1)3
204
To complete the solution, we use the Power Rule for Functions to find ddx(x2+1)3. f′(x)=ex[3(x2+1)2⋅ddx(x2+1)]+ex(x2+1)3Power Rule for Functions=ex[3(x2+1)2⋅2x]+ex(x2+1)3=ex[6x(x2+1)2+(x2+1)3]=ex(x2+1)2[6x+x2+1]=ex(x2+1)2(x2+6x+1)
(b) g is a function raised to a power, so we begin with the Power Rule for Functions. g′(x)=ddx(3x+24x2−5)5=5(3x+24x2−5)4[ddx(3x+24x2−5)]Power Rule for Functions=5(3x+24x2−5)4[(3)(4x2−5)−(3x+2)(8x)(4x2−5)2]Quotient Rule=5(3x+2)4[(12x2−15)−(24x2+16x)](4x2−5)6=5(3x+2)4[−12x2−16x−15](4x2−5)6