"Implicit differentiation with rational exponents Determine the slope..." was answered by our community

High School
Algebra
Algebra I
Exponents and radicals

babeeb0oL

Answered question

2021-09-09

Implicit differentiation with rational exponents Determine the slope of the following curves at the given point.
$\sqrt[3]{x} + \sqrt[3]{y^{4}} = 2; (1, 1)$

Answer & Explanation

Laaibah Pitt

Skilled2021-09-10Added 98 answers

Given curve:
$\sqrt[3]{x} + \sqrt[3]{y^{4}} = 2$
Using implicit differentiation, find $\frac{d y}{d x}$ of the given curve:
First, simplify the given curve:
Apply radical rule: $\sqrt[n]{a} = a^{\frac{1}{n}}$
Apply exponent rule: ${(a^{b})}^{c} = a^{b c}$ , assuming $a \geq 0$
$\sqrt[3]{x} + \sqrt[3]{y^{4}} = 2$
$x^{\frac{1}{3}} + {(y^{4})}^{\frac{1}{3}} = 2$
$x^{\frac{1}{3}} + y^{\frac{4}{3}} = 2$
Now, differentiate both sides with respect to x:
Apply the Power Rule: $\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}$
$\frac{d (x^{\frac{1}{3}} + y^{\frac{4}{3}})}{d x} = \frac{d (2)}{d x}$
$\frac{1}{3} x^{\frac{1}{3} - 1} + \frac{d (y^{\frac{4}{3}})}{d y} \times \frac{d y}{d x} = 0$
$\frac{1}{3} x^{- \frac{2}{3}} + \frac{4}{3} y^{\frac{1}{3}} \times \frac{d y}{d x} = 0$
$\frac{4}{3} y^{\frac{1}{3}} \times \frac{d y}{d x} = - \frac{1}{3} x^{- \frac{2}{3}}$
$\frac{d y}{d x} = \frac{- x^{- \frac{2}{3}}}{4 y^{\frac{1}{3}}}$
$\frac{d y}{d x} = - \frac{1}{4 \sqrt[3]{x^{2}} \sqrt[3]{y}}$
At point (1,1):
Slope $= \frac{d y}{d x} ∣_{}$

Do you have a similar question?Recalculate according to your conditions!

New Questions in Algebra I

Didn't find what you were looking for?