# The equation of the tangent line and derivative of the tangent line at a = 1 where the curve is y = 3 x 2 − x 3 . ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 2.6, Problem 23E
To determine

## To find:The equation of the tangent line and derivative of the tangent line at a=1 where the curve is y=3x2−x3 .

Expert Solution

Value of the function and derivative of the function at a=2 are 3 and 4 respectively.

### Explanation of Solution

Given: y=3x2x3 be any curve at the point (1,2) .

Let y=3x2x3 be any function.

It has to find the value of the equation of the tangent line and derivative of the tangent line at a=1 .

Now definition of the function at a point a .

f'(a)=limh0[f(a+h)f(a)]/hlet  a=1f'(1)==limh0[f(1+h)f(1)]/h

Let x=1 in y=3x2x3 to find

f(1)=3(1)2(1)3=31=2f(1+h)=3(1+h)2(1+h)3=1+2h+h21h33h3h2=2+3hh3

f'(a)=limh0[f(a+h)f(a)]/hf'(1)==limh0[f(1+h)f(1)]/h

f'(1)=limh0[2+3hh22]/hf'(1)=limh0[3hh2]/h=limh0[3h]=3

y=f'(a)(a+h)+f(a)

y=f'(1)(1+h)+f(1)y=3(x1)+2y=3x3+2y=3x1

Hence, the tangent line of the curve is y=3x1 and f'(1)=3 .

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