Podcast
Questions and Answers
Find the derivative of y = ln(t^2)
Find the derivative of y = ln(t^2)
1/t
Find the derivative of y = ln(ln(ln x))
Find the derivative of y = ln(ln(ln x))
1/(ln(ln x) * x)
Find the derivative of y = ln(1/(1-x))
Find the derivative of y = ln(1/(1-x))
1/(1-x)
Find the derivative of y = log_{7}(√(2x-3))
Find the derivative of y = log_{7}(√(2x-3))
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Find the derivative of y = 5^x
Find the derivative of y = 5^x
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Find the derivative of y = 2^x
Find the derivative of y = 2^x
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Use logarithmic differentiation to find the derivative of y = tan(θ)√(2θ + 1)
Use logarithmic differentiation to find the derivative of y = tan(θ)√(2θ + 1)
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Use logarithmic differentiation to find the derivative of y = x(x + 1)
Use logarithmic differentiation to find the derivative of y = x(x + 1)
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Use logarithmic differentiation to find the derivative of y = x sin(x)
Use logarithmic differentiation to find the derivative of y = x sin(x)
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What is the value of the derivative of the function y = (7x^5 - 6x)^{17} when evaluated at x=1?
What is the value of the derivative of the function y = (7x^5 - 6x)^{17} when evaluated at x=1?
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How does the altitude of a triangle relate to the rate of change of its area and base when the altitude is increasing?
How does the altitude of a triangle relate to the rate of change of its area and base when the altitude is increasing?
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How fast is the radius of a spherical balloon changing when gas escapes at 20 cubic feet per minute and the radius is 2 feet?
How fast is the radius of a spherical balloon changing when gas escapes at 20 cubic feet per minute and the radius is 2 feet?
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Study Notes
Derivatives of Logarithmic Functions
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Problem 1: Find the derivative of 𝑦 = ln(t³). Solution involves applying the chain rule and properties of logarithms.
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Problem 2: Find the derivative of 𝑦 = ln(ln(ln x)). This problem requires multiple applications of the chain rule.
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Problem 3: Find the derivative of 𝑦 = ln((1/(1-x))²/(1+x)). This involves simplifying the logarithmic expression before differentiating.
Derivatives of Logarithmic Functions (Continued)
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Problem 4: Find the derivative of y = log₇√(2x - 3). Solution involves changing the base of the logarithm to the natural logarithm (ln) before differentiation.
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Problem 5: Find the derivative of y = 5ˣ. Solution involves rewriting 5ˣ as e^(x ln 5) before differentiating.
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Problem 6: Find the derivative of y = 2ˣ⁴. Solution uses the chain rule and the derivative of exponential functions.
Logarithmic Differentiation
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Logarithmic differentiation simplifies finding derivatives of functions involving products, quotients, and powers.
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Step 1: Take the natural logarithm of both sides of the equation and use logarithmic properties to simplify the expression.
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Step 2: Differentiate implicitly with respect to the independent variable.
Logarithmic Differentiation Examples
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Problem 7: Use logarithmic differentiation to find the derivative of y = tan θ √(2θ + 1).
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Problem 8: Use logarithmic differentiation to find the derivative of y = x(x+1)ˣ.
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Problem 9: Use logarithmic differentiation to find the derivative of y = xˣ sin x.
Derivatives
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Finding derivatives of different functions:
- Power rule: For functions in the form of y = ax^n, the derivative is dy/dx = nax^(n-1)
- Chain rule: For compound functions, the derivative is the product of the derivative of the outer function and the derivative of the inner function.
- Product rule: For functions in the form of y = u(x)v(x), the derivative is dy/dx = u'(x)v(x) + u(x)v'(x)
- Quotient rule: For functions in the form of y = u(x)/v(x), the derivative is dy/dx = (v(x)u'(x) - u(x)v'(x))/v(x)^2
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Trigonometric functions:
- Derivative of sin(x) is cos(x)
- Derivative of cos(x) is -sin(x)
- Derivative of tan(x) is sec^2(x)
- Derivative of cot(x) is -csc^2(x)
- Derivative of sec(x) is sec(x)tan(x)
- Derivative of csc(x) is -csc(x)cot(x)
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Logarithmic functions:
- Derivative of ln(x) is 1/x
- Derivative of log_a(x) is 1/(x ln(a))
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Inverse trigonometric functions:
- Derivative of arcsin(x) is 1/sqrt(1-x^2)
- Derivative of arccos(x) is -1/sqrt(1-x^2)
- Derivative of arctan(x) is 1/(1+x^2)
- Derivative of arccot(x) is -1/(1+x^2)
- Derivative of arcsec(x) is 1/(|x|sqrt(x^2-1))
- Derivative of arccsc(x) is -1/(|x|sqrt(x^2-1))
Second Derivatives
- Finding the second derivative: This involves deriving the first derivative of the function.
- Second derivative application: The second derivative can be used to determine the concavity of the function.
Tangent Lines
- Equation of a tangent line: y - y1 = m(x - x1), where m is the slope of the tangent line and (x1, y1) is a point on the line.
- Finding the slope: The derivative of the function at the point (x1, y1) gives the slope of the tangent line.
Related Rates
- Finding rates of change: Related rates problems involve determining the rate of change of a variable in relation to the rate of change of another variable.
- Key steps: Identify the variables, their rates of change, and the relationship between the variables. Differentiate the relationship between the variables with respect to time. Substitute the known values and solve for the unknown rate of change.
Differentials
- Differentials: Approximating changes in a function using its derivative.
Extrema
- Absolute maximum and minimum: The highest and lowest values of a function within a given interval.
- Finding extrema: The critical points of a function are potential locations for extrema.
Motion
- Velocity: The rate of change of position with respect to time, represented by the derivative of the position function.
- Acceleration: The rate of change of velocity with respect to time, represented by the derivative of the velocity function.
- Total distance traveled: The sum of the distances traveled in both positive and negative directions.
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Description
This quiz covers the derivatives of various logarithmic functions, including natural logarithms and logarithmic differentiation techniques. You will apply the chain rule and properties of logarithms to find derivatives for several complex logarithmic expressions. Test your understanding and improve your calculus skills through these challenging problems.
