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Calculus

Find an where f(x) = x*. equation of the tangent line to the graph of y = f'(x) at x = 3,

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Topics In Analytic Geometry

Section6.2: Introduction To Conics: parabolas

Problem 4ECP: Find an equation of the tangent line to the parabola y=3x2 at the point 1,3.

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Question

Find an
where f(x) = x*.
equation of the tangent line to the graph of y = f'(x) at x = 3,

Expert Solution

Step 1

Given that,

$f (x) = x^{4}$ ...(1)

And y = f'(x)

Step 2

Firstly evaluate y:

Differentiate f(x) with respect to x,

$f' (x) = \frac{d}{d x} (x^{4}) f' (x) = 4 x^{3}$

So $y = 4 x^{3}$

Consider $y = g (x) = 4 x^{3}$ ...(2)

Now evaluate the slope of a tangent line:

Differentiate (2) with respect to x,

$g' (x) = \frac{d}{d x} (4 x^{3}) g' (x) = 12 x^{2}$

So g'(x) at x = 3 is

$\begin{array}{rcl} g' (3) & = & 12 {(3)}^{2} = 108 \end{array}$

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