Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

3. Techniques of Differentiation

The Chain Rule

Calculus

3. Techniques of Differentiation

The Chain Rule

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2:43 minutes

Problem 3.9.38

Textbook Question

15–48. Derivatives Find the derivative of the following functions.
y = 5^3t

Verified step by step guidance

Step 1: Identify the function type. The given function is y = 5^(3t), which is an exponential function where the base is a constant and the exponent is a linear function of t.

Step 2: Recall the derivative rule for exponential functions. If you have a function of the form y = a^(u(t)), where a is a constant and u(t) is a function of t, the derivative is given by y' = a^(u(t)) * ln(a) * u'(t).

Step 3: Apply the derivative rule. In this case, a = 5 and u(t) = 3t. Therefore, the derivative y' = 5^(3t) * ln(5) * (d/dt)(3t).

Step 4: Compute the derivative of the exponent. The derivative of u(t) = 3t with respect to t is simply 3, since the derivative of t is 1 and 3 is a constant multiplier.

Step 5: Combine the results. Substitute the derivative of the exponent back into the formula: y' = 5^(3t) * ln(5) * 3.

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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 a variable. It is a fundamental concept in calculus that allows us to determine how a function behaves as its input changes. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a given point.

Exponential Functions

Exponential functions are mathematical functions of the form y = a^x, where 'a' is a constant and 'x' is the variable. In the context of derivatives, the derivative of an exponential function can be calculated using specific rules, such as the fact that the derivative of a^x is a^x * ln(a). Understanding how to differentiate exponential functions is crucial for solving problems involving them.

Power Rule

The power rule is a basic rule for finding the derivative of functions of the form y = x^n, where 'n' is a real number. According to the power rule, the derivative is given by dy/dx = n*x^(n-1). This rule simplifies the process of differentiation, especially for polynomial and power functions, and is essential for solving a wide range of calculus problems.