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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 3 - DerivativesProblema 70
Capitolo 3, Problema 70

Calculate the derivative of the following functions.
y = e^2x(2x-7)⁵

Guida verificata passo dopo passo
1
Step 1: Identify the structure of the function y = e^{2x}(2x-7)^5. Notice that it is a product of two functions: u(x) = e^{2x} and v(x) = (2x-7)^5.
Step 2: Apply the product rule for differentiation, which states that if y = u(x)v(x), then y' = u'(x)v(x) + u(x)v'(x).
Step 3: Differentiate u(x) = e^{2x}. Use the chain rule: the derivative of e^{2x} is e^{2x} times the derivative of 2x, which is 2. So, u'(x) = 2e^{2x}.
Step 4: Differentiate v(x) = (2x-7)^5. Again, use the chain rule: the derivative is 5(2x-7)^4 times the derivative of the inside function 2x-7, which is 2. So, v'(x) = 10(2x-7)^4.
Step 5: Substitute u(x), u'(x), v(x), and v'(x) into the product rule formula: y' = 2e^{2x}(2x-7)^5 + e^{2x}10(2x-7)^4. Simplify the expression if needed.

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Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the curve at any given point. Derivatives are used to find rates of change, optimize functions, and analyze the behavior of functions.
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Percorso guidato
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Derivatives

Product Rule

The product rule is a formula used to find the derivative of the product of two functions. It states that if you have two functions u(x) and v(x), the derivative of their product is given by u'v + uv'. This rule is essential when differentiating functions that are multiplied together, such as in the given function y = e^(2x)(2x-7)⁵.
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The Product Rule

Chain Rule

The chain rule is a technique for differentiating composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule is particularly useful for functions raised to a power, like (2x-7)⁵ in the given problem.
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Intro to the Chain Rule
Pratica correlata
Domanda del libro di testo

The following equations implicitly define one or more functions.

b. Solve the given equation for y to identify the implicitly defined functions y=f₁(x), y = f₂(x), ….

x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)

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views
Domanda del libro di testo

The following equations implicitly define one or more functions.

c. Use the functions found in part (b) to graph the given equation.

y² = x²(4 − x) / 4 + x (right strophoid)

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Domanda del libro di testo

The following equations implicitly define one or more functions.

a. Find dy/dx using implicit differentiation.

y² = x²(4 − x) / 4 + x (right strophoid)

202
views
Domanda del libro di testo

The following equations implicitly define one or more functions.

a. Find dy/dx using implicit differentiation.

x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)

290
views
Domanda del libro di testo

The following equations implicitly define one or more functions.

b. Solve the given equation for y to identify the implicitly defined functions y=f₁(x), y = f₂(x), ….

y² = x²(4 − x) / 4 + x (right strophoid)

375
views
Domanda del libro di testo

The following equations implicitly define one or more functions.

c. Use the functions found in part (b) to graph the given equation.

x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)

271
views