Textbook Question
5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.
y = √x²+1
3. Techniques of Differentiation
The Chain Rule
4:06 minutesProblem 50
Textbook Question
Calculate the derivative of the following functions.
g(x) = x / e3x
Verified step by step guidance1
Step 1: Identify the function g(x) = \frac{x}{e^{3x}} as a quotient of two functions, where the numerator is f(x) = x and the denominator is h(x) = e^{3x}.
Step 2: Recall the Quotient Rule for derivatives, which states that if you have a function g(x) = \frac{f(x)}{h(x)}, then its derivative g'(x) is given by \frac{f'(x)h(x) - f(x)h'(x)}{(h(x))^2}.
Step 3: Calculate the derivative of the numerator, f'(x). Since f(x) = x, its derivative is f'(x) = 1.
Step 4: Calculate the derivative of the denominator, h'(x). Since h(x) = e^{3x}, use the chain rule to find h'(x) = 3e^{3x}.
Step 5: Substitute f(x), f'(x), h(x), and h'(x) into the Quotient Rule formula to find g'(x) = \frac{1 \cdot e^{3x} - x \cdot 3e^{3x}}{(e^{3x})^2}.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. Derivatives are fundamental in calculus for understanding rates of change and are used in various applications, including optimization and motion analysis.
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Quotient Rule
The quotient rule is a formula used to find the derivative of a function that is the quotient of two other functions. If you have a function g(x) = u(x) / v(x), the derivative g'(x) is given by (u'v - uv') / v², where u' and v' are the derivatives of u and v, respectively. This rule is essential when differentiating functions that are expressed as fractions.
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Exponential Functions
Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where e is Euler's number (approximately 2.71828), and a and b are constants. The derivative of an exponential function is unique because it is proportional to the function itself, meaning that d/dx(e^(bx)) = b * e^(bx). Understanding how to differentiate exponential functions is crucial when they appear in more complex expressions.
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