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EXAMPLE 8Using the Power Rule for Functions with Other Derivative Rules

Find the derivative of:
(a) f(x)=ex(x2+1)3 (b) g(x)=(3x+24x25)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

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To complete the solution, we use the Power Rule for Functions to find ddx(x2+1)3. f(x)=ex[3(x2+1)2ddx(x2+1)]+ex(x2+1)3Power Rule for Functions=ex[3(x2+1)22x]+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+24x25)5=5(3x+24x25)4[ddx(3x+24x25)]Power Rule for Functions=5(3x+24x25)4[(3)(4x25)(3x+2)(8x)(4x25)2]Quotient Rule=5(3x+2)4[(12x215)(24x2+16x)](4x25)6=5(3x+2)4[12x216x15](4x25)6