MATH 1700: Derivative Worksheet
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Questions and Answers

Given $f(x) = e^x \sin x$, what is $f'(x)$?

  • $e^x \sin x$
  • $e^x (\sin x + \cos x)$ (correct)
  • $e^x \cos x$
  • $e^x (\sin x - \cos x)$

If $f(x) = (x^4 + 3x)^{-1}$, what is $f'(x)$?

  • $-(4x^3 + 3)(x^4 + 3x)^{-2}$ (correct)
  • $-(4x^3 + 3)^{-2}$
  • $(4x^3 + 3)(x^4 + 3x)^{-2}$
  • $(x^4 + 3x)^{-2}$

Given $f(x) = \cos^4 x - 2x^2$, determine $f'(x)$.

  • $4 \cos^3 x \sin x - 4x$
  • $- \sin^4 x - 4x$
  • $4 \cos^3 x + 4x$
  • $-4 \cos^3 x \sin x - 4x$ (correct)

Given $f(x) = \ln(xe^{7x})$, find $f'(x)$.

<p>$8$ (B)</p> Signup and view all the answers

If $f(x) = 2x - \sqrt[4]{x}$, what is $f'(x)$?

<p>$2 - (1/4)x^{-3/4}$ (B)</p> Signup and view all the answers

Determine $f'(x)$ for $f(x) = \frac{6}{(3x - \pi)^4}$.

<p>$\frac{-72}{(3x - \pi)^5}$ (D)</p> Signup and view all the answers

Find $f'(x)$ given $f(x) = \arctan(2x)$.

<p>$\frac{2}{1 + 4x^2}$ (D)</p> Signup and view all the answers

Calculate $f'(x)$ for $f(x) = (e^{2x} + e)^2$.

<p>$4e^{2x}(e^{2x} + e)$ (D)</p> Signup and view all the answers

Given $f(x) = \sqrt[3]{x^2 - \sqrt{x^3}}$, which of the following is the correct expression for $f'(x)$?

<p>$f'(x) = \frac{1}{3}(x^2 - \sqrt{x^3})^{-2/3} (2x - \frac{3}{2}x^{1/2})$ (C)</p> Signup and view all the answers

Consider $f(x) = e^x(x^2 + 3)(x^3 + 4)$. Find $f'(x)$.

<p>$e^x(x^2 + 3)(x^3 + 4) + e^x(2x)(x^3 + 4) + e^x(x^2 + 3)(3x^2)$ (C)</p> Signup and view all the answers

Given $f(x) = \frac{\sin x}{\cos x}$, determine $f'(x)$.

<p>$\frac{\cos^2 x + \sin^2 x}{\cos^2 x}$ (C)</p> Signup and view all the answers

If $f(x) = \ln(5x^2 + 9)^3$, what is $f'(x)$?

<p>$\frac{30x}{5x^2 + 9}$ (A)</p> Signup and view all the answers

Given $f(x) = \arcsin(2x)$, find $f'(x)$.

<p>$\frac{2}{\sqrt{1 - 4x^2}}$ (A)</p> Signup and view all the answers

If $3y = xe^{5y}$, find $\frac{dy}{dx}$.

<p>$\frac{e^{5y}}{3 - 5xe^{5y}}$ (A)</p> Signup and view all the answers

If $f(2) = 3$, $f'(2) = -1$, $f'(3) = 7$, $g(2) = -5$, and $g'(2) = 2$, find $(f \cdot g)'(2)$.

<p>11 (C)</p> Signup and view all the answers

Using $f(2) = 3$, $f'(2) = -1$, $f'(3) = 7$, $g(2) = -5$, and $g'(2) = 2$, determine $(\frac{f}{g})'(2)$.

<p>$-\frac{1}{25}$ (B)</p> Signup and view all the answers

Given $f(x) = (x^6 + 1)^5 (4x + 7)^3$, which of the following represents $f'(x)$?

<p>$(x^6 + 1)^5 3(4x + 7)^2 (4) + (4x + 7)^3 5(x^6 + 1)^4 (6x^5)$ (D)</p> Signup and view all the answers

If $f(x) = 6(7x + x^2 + 3)^5 \sqrt{x^2 + 3}$, what is $f'(x)$?

<p>$6(7x + x^2 + 3)^5[7 + (x^2 + 3)^{\frac{-1}{2}} \cdot 2x]$ (C)</p> Signup and view all the answers

Given $f(x) = x^{\frac{2}{3}} + x^{\frac{-1}{2}}$, find $f'(x)$.

<p>$\frac{2}{3}x^{\frac{-1}{3}} - \frac{1}{2}x^{\frac{-3}{2}}$ (C)</p> Signup and view all the answers

If $f(x) = \frac{x^{-1} + x^{-2}}{x - 1}$, what is $f'(x)$?

<p>$\frac{(x - 1)(-x^{-2} - 2x^{-3}) - (x^{-1} + x^{-2})(1)}{(x - 1)^2}$ (A)</p> Signup and view all the answers

What is the derivative of $f(x) = \frac{2x+5}{7x-9}$?

<p>$\frac{(7x - 9)(2) - (2x + 5)(7)}{(7x - 9)^2}$ (D)</p> Signup and view all the answers

Determine $f'(x)$ given that $f(x) = 10 \arctan(2x)$.

<p>$\frac{20}{1 + 4x^2}$ (A)</p> Signup and view all the answers

If $f(x) = (e^{2x} + e)^2$, which of the following is $f'(x)$?

<p>$2(e^{2x} + e)(e^{2x} \cdot 2)$ (D)</p> Signup and view all the answers

Given $f(x) = \pi (xe^x)^{(\pi-1)}$, what is $f'(x)$?

<p>$\pi(\pi-1)(x e^x)^{(\pi-2)} (xe^x + e^x)$ (B)</p> Signup and view all the answers

Given $f(x) = (2x^2 - 5x)^3$, what is $f'(x)$?

<p>$(12x - 15)(2x^2 - 5x)^2$ (B)</p> Signup and view all the answers

If $y = 3\sin(x - 3)$, what is $\frac{dy}{dx}$?

<p>$3\cos(x - 3)$ (A)</p> Signup and view all the answers

Given $g(x) = \sin^2(3x^2)$, find $g'(x)$.

<p>$12x \sin(3x^2) \cos(3x^2)$ (B)</p> Signup and view all the answers

If $f(x) = 3x^3 e^{2x-5}$, what is $f'(x)$?

<p>$9x^2e^{2x-5} + 6x^3e^{2x-5}$ (C)</p> Signup and view all the answers

Given $y = 3x^2 \sqrt[3]{4x^2 - 5x + 1}$, find $\frac{dy}{dx}$.

<p>$6x(4x^2 - 5x + 1)^{1/3} + \frac{x^2(8x - 5)}{(4x^2 - 5x + 1)^{2/3}}$ (A)</p> Signup and view all the answers

If $g(m) = \sin(\cos(m))$, what is $g'(m)$?

<p>$\cos(\cos(m)) \cdot {-\sin(m)}$ (C)</p> Signup and view all the answers

Given $f(x) = (x^3 + 1)^7 \cdot x^3$, which of the following represents the correct application of the product and chain rules to find $f'(x)$ before simplification?

<p>$3x^2 \cdot 7(x^3 + 1)^6 (3x^2) + (x^3 + 1)^7 \cdot 3x^2$ (D)</p> Signup and view all the answers

Given $f(t) = \left( \frac{t^2 + 2}{t^2 - 2} \right)^3$, find $f'(t)$.

<p>$3\left( \frac{t^2 + 2}{t^2 - 2} \right)^2 \cdot \frac{-8t}{(t^2 - 2)^2}$ (B)</p> Signup and view all the answers

Given $f(x) = (x^2 + 2)(5x^2 - 7x)$, which of the following represents $f'(x)$?

<p>$(x^2 + 2)(10x - 7) + (5x^2 - 7x)(2x)$ (B)</p> Signup and view all the answers

If $f(x) = \frac{x}{1 + x^2}$, what is $f'(x)$ before simplification?

<p>$\frac{(1 + x^2)(1) - x(2x)}{(1 + x^2)^2}$ (D)</p> Signup and view all the answers

Given $f(x) = (2 - x)^{\frac{5}{4}} \cdot x^3 $, which expression correctly applies the product rule to find $f'(x)$ before simplification?

<p>$x^3 \cdot \frac{5}{4}(2 - x)^{\frac{1}{4}}(-1) + (2 - x)^{\frac{5}{4}} \cdot 3x^2$ (C)</p> Signup and view all the answers

If $h(x) = e^{\frac{2x^3 - x^2}{3}}$, then $h'(x)$ is:

<p>$\frac{(6x^2 - 2x)}{3} e^{\frac{2x^3 - x^2}{3}}$ (B)</p> Signup and view all the answers

Let $f(x) = 4(\cos x)^3 - 4x$. What is its derivative, $f'(x)$ before simplification?

<p>$12(\cos x)^2(-\sin x) - 4$ (A)</p> Signup and view all the answers

What is the derivative of $f(x) = tan^2(x)$?

<p>$2sec^2(x)tan(x)$ (B)</p> Signup and view all the answers

Suppose $f(x) = \sqrt{x^2 + 8}$. Which of the following represents $f'(x)$?

<p>$\frac{1}{2}(x^2 + 8)^{-\frac{1}{2}} \cdot 2x$ (C)</p> Signup and view all the answers

Determine the derivative of $f(x) = sec(cos(x))$.

<p>$sec(cos(x))tan(cos(x))sin(x)$ (C)</p> Signup and view all the answers

Given $f(x) = \frac{(3x - 1)^2}{(x^2 + 7x)^4}$, what is the correct setup for finding $f'(x)$ using the quotient and chain rules before simplification?

<p>$\frac{(x^2 + 7x)^4 \cdot 2(3x - 1)(3) - (3x - 1)^2 \cdot 4(x^2 + 7x)^3(2x + 7)}{(x^2 + 7x)^8}$ (D)</p> Signup and view all the answers

Find the derivative of $f(x) = sec(x)sin(3x)$.

<p>$3sec(x)cos(3x) + sin(3x)sec(x)tan(x)$ (A)</p> Signup and view all the answers

Consider $f(x) = x^5 - \frac{2}{x^2} + 7x$. What is $f'(x)$ after rewriting $f(x)$ for easier differentiation, but before fully simplifying the derivative?

<p>$5x^4 + 4x^{-3} + 7$ (C)</p> Signup and view all the answers

Given $f(x) = (x-1)^3 / (x(x+3)^4)$, find the general form of $f'(x)$.

<p>$\frac{x(x + 3)^4 3(x - 1)^2 - (x - 1)^3(x(4(x+3)^3) + (x+3)^4)}{(x(x + 3)^4)^2}$ (D)</p> Signup and view all the answers

Let $f(x) = e^x \sin x$. Determine $f'(x)$ before simplification.

<p>$e^x \cos x + e^x \sin x$ (D)</p> Signup and view all the answers

If $f(x) = e^x(x^2 + 3)(x^3 + 4)$, what is $f'(x)$?

<p>$e^x(x^2 + 3)(x^3 + 4) + e^x(2x)(x^3 + 4) + e^x(x^2 + 3)(3x^2)$ (A)</p> Signup and view all the answers

Determine the derivative of $f(x) = arcsin(2^x)$.

<p>$\frac{2^x ln(2)}{\sqrt{1 - 2^{2x}}}$ (C)</p> Signup and view all the answers

Flashcards

Chain Rule Differentiation

A method to find the derivative of composite functions.

Derivative of sin(x)

The derivative of sin(x) is cos(x).

Derivative of cos(x)

The derivative of cos(x) is -sin(x).

Product Rule

A rule to differentiate products of functions: (uv)' = u'v + uv'

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Implicit Differentiation

Used to find dy/dx when y is a function of x implicitly.

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Quotient Rule

A rule to differentiate quotients of functions: (u/v)' = (u'v - uv') / v^2

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Derivative of e^x

The derivative of e raised to x is just e raised to x.

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Nested Chain Rule

Applying the chain rule multiple times to nested composite functions.

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Derivative of (4x^5 - 5x^4)

The derivative of (4x^5 - 5x^4) is found using the power rule. Multiply the coefficient by the exponent and reduce the exponent by one.

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Derivative of (e^x \sin x)

The derivative of (e^x \sin x) requires the product rule. The product rule states: d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x).

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Derivative of ((x^4 + 3x)^{-1})

The derivative of ((x^4 + 3x)^{-1}) requires the chain rule. The chain rule states: d/dx [f(g(x))] = f'(g(x)) * g'(x). Also remember to handle the negative exponent correctly.

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Derivative of (3x^2 (x^3 + 1)^7)

To find the derivative of (3x^2 (x^3 + 1)^7), apply the product rule and the chain rule. First the product rule then chain rule on ((x^3 + 1)^7).

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Derivative of (\cos^4 x - 2x^2)

The derivative of (\cos^4 x - 2x^2) involves the chain rule for (\cos^4 x) and the power rule for (2x^2).

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Derivative of (x / (1 + x^2))

To find the derivative of (x / (1 + x^2)), use the quotient rule: d/dx [u(x)/v(x)] = [v(x)u'(x) - u(x)v'(x)] / [v(x)]^2.

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Derivative of ((x^2 - 1) / x)

To find the derivative of ((x^2 - 1) / x), use the quotient rule: d/dx [u(x)/v(x)] = [v(x)u'(x) - u(x)v'(x)] / [v(x)]^2, or simplify first.

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Derivative of (\ln(xe^{7x}))

First, use properties of logarithms to expand (\ln(xe^{7x})) into (\ln(x) + 7x), then differentiate each term separately.

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Power Rule

The derivative of (x^n) is (nx^{n-1}). Multiply by the original exponent, then reduce the exponent by one.

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Constant Multiple Rule

The derivative of a constant times a function is the constant times the derivative of the function.

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Sum/Difference Rule

The derivative of a sum (or difference) of functions is the sum (or difference) of their derivatives.

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Derivative of (e^x)

The derivative of (e^x) is (e^x).

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Chain Rule

For (f(x) = u(v(x))), (f'(x) = u'(v(x)) \cdot v'(x)). Differentiate the outside, keep the inside, then multiply by the derivative of the inside.

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Derivative of Cosine

The derivative of (\cos(x)) is (-\sin(x)).

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Derivative of Natural Log

The derivative of (\ln(x)) is (\frac{1}{x}).

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Constant Rule

Derivative of a constant is zero.

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Sum Rule

d/dx [f(x) + g(x)] = f'(x) + g'(x)

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Difference Rule

d/dx [f(x) - g(x)] = f'(x) - g'(x)

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What is f '(x)?

f '(x) is the derivative of f(x). It represents the instantaneous rate of change of the function with respect to x.

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Derivative of a Constant

The derivative of a constant is zero.

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Substitution (Derivatives)

Technique used after differentiation, which puts the derivative back in terms of the original variable

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Repeated Chain Rule

Used when differentiating composite functions within other composite functions.

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Power Rule with Chain Rule

(d/dx) [u^n] = n * u^(n-1) * u'

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Derivative of e^u (Chain Rule)

(d/dx) [e^u] = e^u * u'

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Sum Rule in Differentiation

If f(x) = u(x) + v(x), then f'(x) = u'(x) + v'(x).

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Difference Rule in Differentiation

If f(x) = u(x) - v(x), then f'(x) = u'(x) - v'(x)

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Composite Function

A function composed of another function inside: f(g(x))

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Study Notes

  • The document is a derivative worksheet
  • It appears to part of a MATH 1700 course

Derivative Problems

  • Various functions provided require differentiation
  • Problems range in complexity, involving polynomials, trigonometric functions, exponential functions, and logarithms
  • Problems include the product rule, quotient rule, and chain rule

Implicit Differentiation

  • Problems 40-42 involve implicit differentiation, where y is a differentiable function of x
  • Involves finding dy/dx

Problems 43-48

  • Problems 43-48 provide values for differentiable functions f and g, and their derivatives at x=2
  • Problems require finding values related to combinations of these functions and their derivatives, such as (g-f)'(2), (fg)'(2), and (f/g)'(2)

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Description

Practice differentiating various types of functions. Includes problems involving polynomials, trigonometric, exponential, and logarithmic functions. Also covers implicit differentiation, product rule, quotient rule, and chain rule.

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