Question

Find an equation of the tangent line to the curve at the given point.

40. \(y = \sqrt[4]{x} - x,\;\;\;\;\;\;\;\left( {1,0} \right)\)

Short answer

The required equation is \(y = - \frac{3}{4}x + \frac{3}{4}\).

Step-by-step solution

Find the slope of the equation

Determine the derivative \(\frac{{dy}}{{dx}}\) to represent the slope.

\(\begin{aligned}\frac{{dy}}{{dx}} &= \frac{d}{{dx}}\left( {\sqrt[4]{x} - x} \right)\\ &= \frac{d}{{dx}}\left( {{{\left( x \right)}^{1/4}}} \right) - \frac{d}{{dx}}\left( x \right)\\y' &= \frac{1}{4}{x^{\frac{1}{4} - 1}} - 1\\y' &= \frac{1}{4}{x^{ - 3/4}} - 1\end{aligned}\)

Find the slope of the equation at given point

Substitute the given point into the obtained equation of slope to find the slope at particular point.

\(\begin{aligned}y'\left( 1 \right) &= \frac{1}{4}{1^{ - 3/4}} - 1\\ &= \frac{1}{4} - 1\\ &= - \frac{3}{4}\end{aligned}\)

Determine the tangent line equation

The equation of tangent line is given by \(y - {y_1} = m\left( {x - {x_1}} \right)\), where \(\left( {{x_1},{y_1}} \right)\) is the given pint and \(m\) is the slope at given point.

Substitute the values into the equation to find the equation of tangent line at the point \(\left( {1,0} \right)\).

\(\begin{aligned}y - 0 = - \frac{3}{4}\left( {x - 1} \right)\\y = - \frac{3}{4}x + \frac{3}{4}\end{aligned}\)

Thus, the required equation is \(y = - \frac{3}{4}x + \frac{3}{4}\).