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Chapter 16: Problem 4

Find the derivative of the following functions: (a) \(y=\mathrm{e}^{2 x} x^{3} \sin 3 x \cos 2 x\) (b) \(y=(x+\sin x)^{7}\) (c) \(H=\ln \left(t^{2}+3 t-9\right)\) (d) \(V(r)=\frac{1}{\ln r}\) (e) \(M(b)=\ln b+\ln (b+1)\)

Short Answer

Expert verified
Question: Find the derivatives of the following functions: (a) \(y = \mathrm{e}^{2 x}\cdot x^{3} \cdot \sin(3 x )\cdot \cos(2x)\) (b) \(y = (x+\sin x)^{7}\) (c) \(H = \ln(t^2+3t-9)\) (d) \(V(r) = -\frac{\ln r}{(1)}\) (e) \(M(b) = \ln b + \ln(b+1)\) Answer: (a) \(y' = 2\mathrm{e}^{2 x}\cdot x^{3} \cdot \sin(3 x )\cdot \cos(2x) + \mathrm{e}^{2 x}\cdot 3x^{2} \cdot \sin(3 x )\cdot \cos(2x) + \mathrm{e}^{2 x}\cdot x^{3} \cdot 3\cos(3 x) \cdot \cos(2x) + \mathrm{e}^{2 x}\cdot x^{3} \cdot \sin(3 x )\cdot (-2\sin(2x))\) (b) \(y' = 7(x+\sin x)^{6}\cdot (1+\cos x)\) (c) \(H' = \frac{2t + 3}{t^2+3t-9}\) (d) \(V'(r) = \frac{-1}{r(\ln r)^2}\) (e) \(M'(b) = \frac{1}{b} + \frac{1}{b+1}\)

Step by step solution

01

Identify the rules necessary for differentiation

For the first function, we need to use the product rule and the chain rule, since the function is a product of various kinds of functions.
02

Apply the product rule and chain rule to find the derivative

We will differentiate each term one by one and combine them together. Using the product rule: \(\frac{d}{d x}(uv) = u'v + uv'\) \(y' = (\mathrm{e}^{2 x})'\cdot x^{3} \cdot \sin(3 x )\cdot \cos(2x) + \mathrm{e}^{2 x}\cdot (x^{3})' \cdot \sin(3 x )\cdot \cos(2x) + \mathrm{e}^{2 x}\cdot x^{3} \cdot (\sin(3 x))' \cdot \cos(2x) + \mathrm{e}^{2 x}\cdot x^{3} \cdot \sin(3 x )\cdot (\cos(2x))'\) Now, we apply the chain rule for each derivative found above: \(y' = 2\mathrm{e}^{2 x}\cdot x^{3} \cdot \sin(3 x )\cdot \cos(2x) + \mathrm{e}^{2 x}\cdot 3x^{2} \cdot \sin(3 x )\cdot \cos(2x) + \mathrm{e}^{2 x}\cdot x^{3} \cdot 3\cos(3 x) \cdot \cos(2x) + \mathrm{e}^{2 x}\cdot x^{3} \cdot \sin(3 x )\cdot (-2\sin(2x))\) (b)
03

Identify the rule necessary for differentiation

We will use the chain rule and power rule since we have a combination of polynomial and trigonometric functions raised to a power.
04

Apply the chain rule to find the derivative

\(y' = (7)(x+\sin x)^{6}\cdot (1+\cos x)\) (c)
05

Identify the rule necessary for differentiation

We will use the chain rule for this function since it involves the natural logarithm and a polynomial inside it.
06

Apply the chain rule to find the derivative

\(H' = \frac{1}{t^2+3t-9}\cdot (2t + 3)\) (d)
07

Identify the rule necessary for differentiation

We will use the quotient rule for this function since it involves a fraction.
08

Apply the quotient rule to find the derivative

Quotient rule: \(\frac{d}{d x}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}\) \(V'(r) = \frac{(-1)(\ln r)'(1)}{(\ln r)^2} = \frac{-1}{r(\ln r)^2} \) (e)
09

Simplify the function

Combine the logarithmic expressions using the property \(\ln a + \ln b = \ln (ab)\): \(M(b) = \ln (b(b+1))\)
10

Apply the chain rule to find the derivative

\(M'(b) = \frac{1}{b(b+1)}\cdot (b+1+1b') = \frac{1+b'}{b(b+1)} = \frac{1}{b} + \frac{1}{b+1}\)

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Most popular questions from this chapter

Find \(\frac{\mathrm{dy}}{\mathrm{d} x}\) where \(y\) is given by (a) \(x \cos x\) (b) \(x e^{x^{k}}\) (c) \(\sin x \cos 2 x\) (d) \(x^{3} \mathrm{e}^{2 x}\) (e) \(x^{4} \sin 2 x\)

Find \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) given \(y=\mathrm{e}^{\left(x^{2}\right)}\).

Find the equation of the tangent to \(y=\frac{1}{x^{2}}\) where \(x=-1 .\)

Use parametric differentiation to find \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) given (a) \(x=1+t, y=2+3 t+t^{2}\) (b) \(x=\sin t, y=\cos t\) (c) \(x=t^{2}, y=t^{3}\) (d) \(x=\mathrm{e}^{t}, y=\mathrm{e}^{t}+t\) (e) \(x=\sqrt{t}, y=1+\ln t\)

State the range of values of \(x\) for which each of the following functions is (i) concave up (ii) concave down: (a) \(y=\frac{x^{3}}{6}-\frac{5 x^{2}}{2}+3 x-9\) (b) \(y=3+x+\frac{x^{2}}{2}-\frac{x^{4}}{12}\) (c) \(y=\mathrm{e}^{x}-100 x-100\) (d) \(y=(x-1)^{4}\) (e) \(y=\frac{x^{2}}{2} \ln x-\frac{x^{2}}{4}\)
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