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Chapter 7: Problem 21

Find the derivative of the function. $$ y=4 x^{-2}+2 x^{2} $$

Short Answer

Expert verified
The derivative of the function \(y=4x^{-2}+2x^{2}\) is \(-8x^{-3} + 4x\).

Step by step solution

01

Identify the Terms of the Function

The given function is \(y=4x^{-2}+2x^{2}\). Here, there are two terms, \(4x^{-2}\) and \(2x^{2}\). From the power rule of differentiation, the derivative of \(x^n\), where n is any real number, is \(nx^{n-1}\). Therefore, for each term, the derivative can be found by multiplying the entire term by the power of x, and then subtracting one from the power.
02

Find the Derivative of the First term

The first term is \(4x^{-2}\). When differentiating, the derivative becomes \(-2*4*x^{-2-1}\) which simplifies to \(-8x^{-3}\).
03

Find the Derivative of the Second Term

The second term is \(2x^{2}\). When differentiating, the derivative becomes \(2*2*x^{2-1}\) which simplifies to \(4x\).
04

Combine the Derivatives of the Terms

Finally, to find the derivative of the entire function, the derivatives of the individual terms are combined. Therefore, the derivative of the given function, \(y=4x^{-2} + 2x^{2}\), is \(-8x^{-3} + 4x\).

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Most popular questions from this chapter

Use the General Power Rule to find the derivative of the function. $$ h(x)=\left(6 x-x^{3}\right)^{2} $$

The value \(V\) of a machine \(t\) years after it is purchased is inversely proportional to the square root of \(t+1\). The initial value of the machine is (a) Write \(V\) as a function of \(t\). (b) Find the rate of depreciation when \(t=1\). (c) Find the rate of depreciation when \(t=3\).

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=t^{2} \sqrt{t-2} $$

Identify the inside function, \(u=g(x)\), and the outside function, \(y=f(u)\). $$ y=f(g(x)) \quad u=g(x) \quad y=f(u) $$ $$ y=\left(4-x^{2}\right)^{-1} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ h(t)=\frac{t+2}{t^{2}+5 t+6} $$
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