Textbook Question
75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = (x+1)^3/2(x-4)^5/2 / (5x+3)^2/3
6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Logarithmic Differentiation
4:14 minutesProblem 3.9.92
Textbook Question
Calculate the derivative of the following functions (i) using the fact that bx = exIn b and (ii) using logarithmic differentiation. Verify that both answers are the same.
y = (x²+1)x
Verified step by step guidance1
Step 1: For part (i), rewrite the function y = (x²+1)^x using the property b^x = e^(x ln b). This gives y = e^(x ln(x²+1)).
Step 2: Differentiate y = e^(x ln(x²+1)) with respect to x using the chain rule. The derivative of e^u with respect to x is e^u * du/dx, where u = x ln(x²+1).
Step 3: Find du/dx where u = x ln(x²+1). Use the product rule for differentiation: if u = v * w, then du/dx = v' * w + v * w'. Here, v = x and w = ln(x²+1).
Step 4: For part (ii), use logarithmic differentiation. Take the natural logarithm of both sides: ln(y) = x ln(x²+1). Differentiate both sides with respect to x. The left side becomes (1/y) * dy/dx, and the right side is differentiated using the product rule.
Step 5: Solve for dy/dx in both methods and verify that the expressions obtained are equivalent, confirming that both methods yield the same derivative.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
4mWas this helpful?
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 a fundamental concept in calculus, representing the slope of the tangent line to the curve of the function at any given point. The derivative can be calculated using various rules and techniques, such as the power rule, product rule, and chain rule.
Recommended video:
05:44
Derivatives
Exponential Functions
Exponential functions are mathematical functions of the form f(x) = b^x, where b is a positive constant. They exhibit rapid growth or decay and are characterized by their unique property that the rate of change of the function is proportional to its current value. The expression b^x can be rewritten using the natural exponential function as e^(x ln b), which simplifies differentiation.
Recommended video:
6:13Exponential Functions
Logarithmic Differentiation
Logarithmic differentiation is a technique used to differentiate functions that are products or quotients of variables raised to variable powers. By taking the natural logarithm of both sides of the equation, the differentiation process becomes simpler, allowing the use of properties of logarithms to separate variables. This method is particularly useful for functions like y = (x² + 1)^x, where both the base and the exponent are variable.
Recommended video:
06:30Logarithmic Differentiation
6:30mWatch next
Master Logarithmic Differentiation with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice