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Chapter 3: Problem 11

Find the derivative of the following functions. $$h(x)=(x-1)\left(x^{3}+x^{2}+x+1\right)$$

Short Answer

Expert verified
Answer: The derivative of the function $$h(x)$$ is $$h'(x) = 4x^{3}$$.

Step by step solution

01

Identify the functions

We have the function $$h(x)=(x-1)\left(x^{3}+x^{2}+x+1\right)$$. We can see that it's a product of two functions, let's call them $$f(x)$$ and $$g(x)$$: $$f(x) = x-1$$ $$g(x) = x^{3}+x^{2}+x+1$$
02

Apply the Product Rule

The product rule for differentiation states that, if $$h(x) = f(x)g(x)$$, then the derivative of $$h(x)$$ is: $$h'(x) = f'(x)g(x) + f(x)g'(x)$$ Now, we need to find the derivatives of $$f(x)$$ and $$g(x)$$.
03

Find f'(x)

To find the derivative of $$f(x) = x-1$$, we have: $$f'(x) = \frac{d}{dx}(x-1) = 1$$
04

Find g'(x)

To find the derivative of $$g(x) = x^{3}+x^{2}+x+1$$, we have: $$g'(x) = \frac{d}{dx}(x^{3}+x^{2}+x+1) = 3x^{2}+2x+1$$
05

Substitute into Product Rule

Now that we have $$f'(x)$$ and $$g'(x)$$, we can substitute them into the product rule formula along with the original functions $$f(x)$$ and $$g(x)$$: $$h'(x) = f'(x)g(x) + f(x)g'(x) = (1)(x^{3}+x^{2}+x+1) + (x-1)(3x^{2}+2x+1)$$
06

Simplify

Now we can simplify the expression to find the derivative of $$h(x)$$: \begin{align*} h'(x) &= x^{3}+x^{2}+x+1 + 3x^{3}+2x^{2}+x - 3x^{2}-2x-1\\ &= x^{3} + 3x^{3} + x^{2} + 2x^{2} - 3x^{2} + x - 2x + 1 -1 \\ &= 4x^{3}+0 = 4x^{3} \end{align*} So, the derivative of $$h(x)$$ is: $$h'(x) = 4x^{3}$$

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Most popular questions from this chapter

Derivatives and inverse functions Find the slope of the curve \(y=f^{-1}(x)\) at (4,7) if the slope of the curve \(y=f(x)\) at (7,4) is \(\frac{2}{3}\)

Given the function \(f,\) find the slope of the line tangent to the graph of \(f^{-1}\) at the specified point on the graph of \(f^{-1}\) . $$f(x)=(x+2)^{2} ;(36,4)$$

Prove the following identities and give the values of \(x\) for which they are true. $$\tan \left(2 \tan ^{-1} x\right)=\frac{2 x}{1-x^{2}}$$

Given the function \(f,\) find the slope of the line tangent to the graph of \(f^{-1}\) at the specified point on the graph of $$f(x)=x^{3} ;(8,2)$$

Special Product Rule In general, the derivative of a product is not the product of the derivatives. Find nonconstant functions \(f\) and \(g\) such that the derivative of \(f g\) equals \(f^{\prime} g^{\prime}\).
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