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More Lessons for Calculus

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### The Chain Rule

The following figure gives the Chain Rule that is used to find the derivative of composite functions. Scroll down the page for more examples and solutions.

### Chain Rule: The General Power Rule

The general power rule is a special case of the chain rule. It is useful when finding the derivative of a function that is raised to the nth power. The general power rule states that this derivative is n times the function raised to the (n-1)th power times the derivative of the function.

This tutorial presents the chain rule and a specialized version called the generalized power rule. Several examples are demonstrated.

Errata: at (9:00) the question was changed from x^{2} to x^{4}
### Chain Rule: The General Exponential Rule

The exponential rule is a special case of the chain rule. It is useful when finding the derivative of e raised to the power of a function. The exponential rule states that this derivative is e to the power of the function times the derivative of the function.

Derivatives of Exponential Functions. Just some examples of finding derivatives of functions involving exponentials.### Chain Rule: The General Logarithm Rule

The logarithm rule is a special case of the chain rule. It is useful when finding the derivative of the natural logarithm of a function. The logarithm rule states that this derivative is 1 divided by the function times the derivative of the function.
### Examples using the Chain Rule

More Lessons for Calculus

Math Worksheets

In Leibniz notation, if y = f(u) and u = g(x) are both differentiable functions, then

Note: In the Chain Rule, we work from the outside to the inside. We differentiate the outer function and then we multiply with the derivative of the inner function.

**Example:**

Find the derivatives of each of the following

**Solution:**

**Example**:

Differentiate y = (2x + 1)^{5}(x^{3} – x +1)^{4}

**Solution:**

In this example, we use the Product Rule before using the Chain Rule.

This tutorial presents the chain rule and a specialized version called the generalized power rule. Several examples are demonstrated.

Errata: at (9:00) the question was changed from x

Derivatives of Exponential Functions. Just some examples of finding derivatives of functions involving exponentials.

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