"For f(x)=(x2+3x−10)3, what is the equation of the..." solved and explained

High School
Calculus and Analysis
Calculus 1
Derivatives

ennuvolat4gv

Answered question

2022-02-16

For

f (x) = {(x^{2} + 3 x - 10)}^{3}

, what is the equation of the line tangent to x=5 and at what point is the tangent line horizontal?

Answer & Explanation

Josef Beil

Beginner2022-02-17Added 12 answers

We start by differentiating f(x), using the chain rule.
Letting

y = u^{3}

and

u = x^{2} + 3 x - 10

. Then

\frac{d y}{d u} = 3 u^{2}

and (du)/dx=2x+3
Then:

\frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}

\frac{d y}{d x} = 3 u^{2} \times 2 x + 3

\frac{d y}{d x} = (6 x + 9) {(x^{2} + 3 x - 10)}^{2}

The tangent will be horizontal if its slope is 0. The derivative represents the rate of change of the function at any given point in its domain.
We set the derivative to 0 and solve.

0 = (6 x + 9) {(x^{2} + 3 x - 10)}^{2}

0=6x+9 and

0 = x^{2} + 3 x - 10

x = - \frac{9}{6}, 2, - 5 = - \frac{3}{2}, 2, - 5

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Calculus 1

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function
$f (x, y) = x^{3} - 6 x y + 8 y^{3}$
$\frac{1}{\sec (x)}$ in derivative?
What is the derivative of $\ln (x + 1)$ ?
What is the limit of $e^{- x}$ as $x \to \infty$ ?
What is the derivative of $f (x) = 5^{\ln x}$ ?
What is the derivative of $e^{- 2 x}$ ?
How to find $lim \frac{e^{t} - 1}{t}$ as $t \to 0$ using l'Hospital's Rule?
What is the integral of $\sqrt{9 - x^{2}}$ ?
What is the derivative of $f (x) = \ln [x^{9} {(x + 3)}^{6} {(x^{2} + 7)}^{5}]$ ?
What Is the common difference or common ratio of the sequence 2, 5, 8, 11...?
How to find the derivative of $y = e^{5 x}$ ?
How to evaluate the limit $\frac{\sin (5 x)}{x}$ as x approaches 0?
How to find derivatives of parametric functions?
What is the antiderivative of $- 5 e^{x - 1}$ ?
How to evaluate: indefinite integral $\frac{1 + x}{1 + x^{2}} d x$ ?

Didn't find what you were looking for?