Textbook Question
Suppose the line tangent to the graph of f at x=2 is y=4x+1 and suppose y=3x−2 is the line tangent to the graph of g at x=2. Find an equation of the line tangent to the following curves at x=2.
y = f(x)g(x)
3. Techniques of Differentiation
Product and Quotient Rules
4:11 minutesProblem 3.4.23
Textbook Question
Derivatives Find and simplify the derivative of the following functions.
f(t) = t⁵/³e^t
Verified step by step guidance1
Step 1: Identify the function f(t) = t^{5/3} e^t as a product of two functions: u(t) = t^{5/3} and v(t) = e^t.
Step 2: Apply the product rule for derivatives, which states that if you have a function h(t) = u(t)v(t), then h'(t) = u'(t)v(t) + u(t)v'(t).
Step 3: Differentiate u(t) = t^{5/3} using the power rule. The power rule states that if u(t) = t^n, then u'(t) = n t^{n-1}. So, u'(t) = \frac{5}{3} t^{\frac{5}{3} - 1}.
Step 4: Differentiate v(t) = e^t. The derivative of e^t with respect to t is simply e^t, so v'(t) = e^t.
Step 5: Substitute u(t), u'(t), v(t), and v'(t) into the product rule formula: f'(t) = u'(t)v(t) + u(t)v'(t) = \frac{5}{3} t^{\frac{2}{3}} e^t + t^{\frac{5}{3}} e^t.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.
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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(t) and v(t), the derivative of their product is given by u'v + uv'. This rule is essential when differentiating functions that are products of simpler functions, such as polynomials and exponentials.
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Exponential Functions
Exponential functions are functions of the form f(t) = a * e^(kt), where 'e' is the base of natural logarithms, and 'a' and 'k' are constants. The derivative of an exponential function is unique because it is proportional to the function itself, making it straightforward to differentiate. Understanding how to differentiate exponential functions is crucial when they are part of more complex expressions.
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