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Chapter 3: Problem 7

Find the derivative of the following functions. $$f(x)=3 x^{4}\left(2 x^{2}-1\right)$$

Short Answer

Expert verified
Question: Find the derivative of the function $$f(x) = 3x^4(2x^2 - 1)$$. Answer: The derivative of the given function is $$f'(x) = 36x^5 - 12x^3$$.

Step by step solution

01

Recall the Differentiation Rules

We need to recall two differentiation rules to solve this problem: 1. Power Rule: The derivative of $$x^n$$ with respect to x is $$nx^{n-1}$$. 2. Product Rule: The derivative of the product of two functions $$u(x)$$ and $$v(x)$$ is $$\frac{d(uv)}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}$$.
02

Identify Functions u and v

Write given function $$f(x)$$ as a product of two functions, u(x) and v(x): $$f(x) = u(x)v(x)$$ Here, $$u(x) = 3x^4$$ and $$v(x) = (2x^2 - 1)$$.
03

Find the Derivative of u(x) and v(x)

Using the power rule, find the derivative of each function: $$\frac{du}{dx} = \frac{d(3x^4)}{dx} = 12x^3$$ $$\frac{dv}{dx} = \frac{d(2x^2 - 1)}{dx} = 4x$$
04

Apply the Product Rule

Apply the product rule to find the derivative of $$f(x)$$: $$f'(x) = u\frac{dv}{dx} + v\frac{du}{dx}$$ $$f'(x) = (3x^4)(4x) + (2x^2 - 1)(12x^3)$$
05

Simplify the Derivative

Simplify the derivative: $$f'(x) = 12x^5 + 24x^5 - 12x^3$$ Combine like terms: $$f'(x) = 36x^5 - 12x^3$$ So, the derivative of the given function is: $$f'(x) = 36x^5 - 12x^3$$

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Most popular questions from this chapter

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

One of the Leibniz Rules One of several Leibniz Rules in calculus deals with higher-order derivatives of products. Let \((f g)^{(n)}\) denote the \(n\) th derivative of the product \(f g,\) for \(n \geq 1\) a. Prove that \((f g)^{(2)}=f^{\prime \prime} g+2 f^{\prime} g^{\prime}+f g^{\prime \prime}\) b. Prove that, in general,$$(f g)^{(n)}=\sum_{k=0}^{n}\left(\begin{array}{l} n \\\k\end{array}\right) f^{(k)} g^{(n-k)}$$ where \(\left(\begin{array}{l}n \\\ k\end{array}\right)=\frac{n !}{k !(n-k) !}\) are the binomial coefficients. c. Compare the result of (b) to the expansion of \((a+b)^{n}\).

The hands of the clock in the tower of the Houses of Parliament in London are approximately \(3 \mathrm{m}\) and \(2.5 \mathrm{m}\) in length. How fast is the distance between the tips of the hands changing at 9:00? (Hint: Use the Law of cosines.)

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals. . A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas \(y=c x^{2}\) form orthogonal trajectories with the family of ellipses \(x^{2}+2 y^{2}=k,\) where \(c\) and \(k\) are constants (see figure). Use implicit differentiation if needed to find \(d y / d x\) for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. \(y=m x ; x^{2}+y^{2}=a^{2},\) where \(m\) and \(a\) are constants

Derivatives and inverse functions Suppose the slope of the curve \(y=f^{-1}(x)\) at (4,7) is \(\frac{4}{5}\) Find \(f^{\prime}(7)\)
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