Question

Differentiate

\(y = \left( {{\bf{10}}{x^{\bf{2}}} + {\bf{7}}x - {\bf{2}}} \right)\left( {{\bf{2}} - {x^{\bf{2}}}} \right)\)

Short answer

The derivative of y is \( - 40{x^3} - 21{x^2} + 44x + 14\).

Step-by-step solution

Step 1:Find the derivative of f using the product rule

The equation for product rule is \(\left( {fg} \right)' = fg' + gf'\).

Apply product rule for the function \(y\left( x \right) = \left( {10{x^2} + 7x - 2} \right)\left( {2 - {x^2}} \right)\).

\(\begin{aligned}y'\left( x \right) &= \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {\left( {10{x^2} + 7x - 2} \right)\left( {2 - {x^2}} \right)} \right)\\ &= \left( {10{x^2} + 7x - 2} \right)\frac{{\rm{d}}}{{{\rm{d}}x}}\left( {2 - {x^2}} \right) + \left( {2 - {x^2}} \right)\frac{{\rm{d}}}{{{\rm{d}}x}}\left( {10{x^2} + 7x - 2} \right)\end{aligned}\)

Differentiate the equation in step 1

The derivative of y can be obtained as,

\(\begin{aligned}y'\left( x \right) &= \left( {10{x^2} + 7x - 2} \right)\frac{{\rm{d}}}{{{\rm{d}}x}}\left( {2 - {x^2}} \right) + \left( {2 - {x^2}} \right)\frac{{\rm{d}}}{{{\rm{d}}x}}\left( {10{x^2} + 7x - 2} \right)\\ &= \left( {10{x^2} + 7x - 2} \right)\left( { - 2x} \right) + \left( {2 - {x^2}} \right)\left( {20x + 7} \right)\\ &= - 20{x^3} - 14{x^2} + 4x + 40x + 14 - 20{x^3} - 7{x^2}\\ &= - 40{x^3} - 21{x^2} + 44x + 14\end{aligned}\)

Thus, the derivative of y is \( - 40{x^3} - 21{x^2} + 44x + 14\).