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Chapter 4: Problem 7

Write down the derivative of (a) \(y=e^{6 x}\) (b) \(y=e^{-342 x}\) (c) \(y=2 e^{-x}+4 e^{x}\) (d) \(y=10 e^{4 x}-2 x^{2}+7\)

Short Answer

Expert verified
a) \( 6e^{6x} \ \) b) \( -342e^{-342x} \ \) c) \( -2e^{-x} + 4e^{x} \ \) d) \( 40e^{4x} - 4x \).

Step by step solution

01

Derivative of the exponential function in part (a)

To find the derivative of the function \( y = e^{6x} \), use the chain rule. The chain rule states that if \( y = e^{u} \) and \( u \) is a function of \( x \), then the derivative \( \frac{dy}{dx} = e^{u} \frac{du}{dx} \). Here, \( u = 6x \) and \( \frac{du}{dx} = 6 \). Therefore, the derivative of \( y = e^{6x} \) is \( \frac{dy}{dx} = 6e^{6x} \).
02

Derivative of the exponential function in part (b)

Similar to step 1, for \( y = e^{-342x} \), we use the chain rule. Here, \( u = -342x \) and \( \frac{du}{dx} = -342 \). Hence, the derivative of \( y = e^{-342x} \) is \( \frac{dy}{dx} = -342e^{-342x} \).
03

Derivative of the composite function in part (c)

For the function \( y = 2e^{-x} + 4e^{x} \), find the derivative of each term separately. For \( 2e^{-x} \), using the chain rule, \( u = -x \) and \( \frac{du}{dx} = -1 \), giving \( \frac{d}{dx}(2e^{-x}) = -2e^{-x} \). For \( 4e^{x} \), \( u = x \) and \( \frac{du}{dx} = 1 \), so \( \frac{d}{dx}(4e^{x}) = 4e^{x} \). Adding them together gives \( \frac{dy}{dx} = -2e^{-x} + 4e^{x} \).
04

Derivative of the polynomial and exponential function in part (d)

For the function \( y = 10e^{4x} - 2x^{2} + 7 \), differentiate each term separately. For \( 10e^{4x} \), \( u = 4x \) and \( \frac{du}{dx} = 4 \), giving \( \frac{d}{dx}(10e^{4x}) = 40e^{4x} \). For \( -2x^{2} \), the power rule \( \frac{d}{dx}(x^{n}) = nx^{n-1} \) gives \( \frac{d}{dx}(-2x^{2}) = -4x \). The constant term \( 7 \) has a derivative of \( 0 \). Combining these, \( \frac{dy}{dx} = 40e^{4x} - 4x \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

chain rule
The chain rule is a fundamental technique in calculus for finding the derivative of a composite function. It helps us differentiate functions where one function is inside another. The general form is: if you have a function y = f(g(x)), then the derivative of y with respect to x is given by dy/dx = f'(g(x)) * g'(x). In simpler terms:
  • Differentiate the outer function, keeping the inner function unchanged.
  • Multiply by the derivative of the inner function.
For instance, in the exercise, to find the derivative of y = e^{6x}, we let u = 6x and apply the chain rule. The outer function is e^u, whose derivative is e^u, and the inner function's derivative is 6. So, dy/dx = e^{6x} * 6 = 6e^{6x}. Breaking your work into these steps makes it easier to apply the chain rule accurately.
exponential differentiation
Differentiating exponential functions follows precise rules, especially when the exponent contains a variable. For any exponential function of the form y = e^{u(x)}, where u(x) is some function of x, the derivative is found using the chain rule: dy/dx = e^{u(x)} * du/dx. Let's see this in our original problems:
  • For y = e{6x}, using the chain rule gives us 6e{6x}, as shown above.
  • For y = e^{-342x}, similarly, u = -342x and du/dx = -342, so dy/dx = -342e^{-342x}.
Ensuring mastery of these differentiation steps for exponentials helps simplify and verify your calculus work.
polynomial differentiation
Polynomials are simple to differentiate using well-known rules. Each term of the polynomial is handled individually. For a term in the form of ax^{n}, where a is a constant and n is a natural number, use the power rule for differentiation: the derivative of ax^{n} is anx^{n-1}. For example, in the exercise, we encountered y = 10e^{4x} - 2x^{2} + 7. To differentiate:
  • The term -2x^{2} uses the power rule: d/dx(-2x^{2}) = -4x.
  • The derivative of the constant term 7 is 0.
These steps lead us to the combined result dy/dx = 40e^{4x} - 4x, verifying the polynomial differentiation for each term.
power rule
The power rule is one of the simplest and most frequently used differentiation rules. It states: if you have a function y = x^{n}, its derivative is y' = nx^{n-1}. Here鈥檚 how it works:
  • Multiply the exponent by the coefficient.
  • Reduce the exponent by 1.
This rule applies to any real number n. For example, considering our function in part (d), we have y = -2x^{2}. Using the power rule: d/dx(-2x^{2}) = -4x. Always remember, the power rule can only be used where the variable base is raised to a constant exponent.

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