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Chapter 3: Problem 18

Find \(d y / d x\) for the following functions. $$y=5 x^{2}+\cos x$$

Short Answer

Expert verified
Answer: The derivative of the function is \(\frac{dy}{dx} = 10x - \sin{x}\).

Step by step solution

01

Differentiate the power function

To differentiate the power function, \(5x^2\), with respect to x, we can apply the power rule which is: \((ax^n)' = anx^{n-1}\). Here, \(a = 5\) and \(n = 2\). So, the derivative will be: \((5x^2)' = 5\cdot2x^{(2-1)}\).
02

Differentiate the trigonometric function

To differentiate the trigonometric function, \(\cos{x}\), with respect to x, we should know that the derivative of \(\cos{x}\) is \(-\sin{x}\). So, the derivative will be: \((\cos{x})' = -\sin{x}\).
03

Combine the derivatives

Now, we can combine the derivatives of both functions. As the function \(y\) is the sum of these two functions, the derivative will be the sum of their derivatives. So, we have: $$\frac{dy}{dx}=(5x^2)'+(\cos{x})'=10x-\sin{x}$$

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Most popular questions from this chapter

Product Rule for three functions Assume that \(f, g,\) and \(h\) are differentiable at \(x\) a. Use the Product Rule (twice) to find a formula for \(\frac{d}{d x}(f(x) g(x) h(x))\) b. Use the formula in (a) to find \(\frac{d}{d x}\left(e^{2 x}(x-1)(x+3)\right)\)

Compute the derivative of the following functions. \(h(x)=\frac{(x+1)}{x^{2} e^{x}}\)

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

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The derivative \(\frac{d}{d x}\left(e^{5}\right)\) equals \(5 \cdot e^{4}\) b. The Quotient Rule must be used to evaluate \(\frac{d}{d x}\left(\frac{x^{2}+3 x+2}{x}\right)\) c. \(\frac{d}{d x}\left(\frac{1}{x^{5}}\right)=\frac{1}{5 x^{4}}\) d. \(\frac{d^{n}}{d x^{n}}\left(e^{3 x}\right)=3^{n} \cdot e^{3 x},\) for any integer \(n \geq 1\)

Visualizing tangent and normal lines a. Determine an equation of the tangent line and normal line at the given point \(\left(x_{0}, y_{0}\right)\) on the following curves. (See instructions for Exercises \(63-68 .)\) b. Graph the tangent and normal lines on the given graph. \(\left(x^{2}+y^{2}\right)^{2}=\frac{25}{3}\left(x^{2}-y^{2}\right);\) \(\left(x_{0}, y_{0}\right)=(2,-1)\) (lemniscate of Bernoulli)
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