Product rule, Chain rule and Higher derivatives

Bridge Course, August 2012

**Product Rule**

** Intuition**: The derivative of a product is not the product of the derivatives. That is, differentiation does not distribute over multiplication. To ﬁnd the derivative of a product expression, we need to use product rule.

** Deﬁnition**: In calculus, the product rule is a formula used to ﬁnd the derivatives of products of functions. It is deﬁned as

or written in another format:

There is another rule for quotients.That is quotient rule and it is deﬁned as

**Examples MAXIMA CODE**

1. ** ** Find .

Consider and

and and hence,

diff((x ^ 2 + 2 * x + 3) * (x ^ 3 + 2 * x − 2), x, 1)

2. Find .

Consider and

and and hence,

diff(%e ^ (3 * x) * tan(x), x, 1)

**Exercises**

Find the derivative of the following function

diff((2 * x − 1) * (x + 1) ^ 2, x, 1)

diff(%e ^ x * cos(x) * sin(x), x, 1)

diff(x * log(x), x, 1)

diff(cos(x)/sin(x), x, 1)

diff(%e ^ x * tan(x)/x, x, 1)

**Chain Rule**

**Intuition**: The chain rule states that if we have a function of the form *y(u(x))* (i.e. *y* can be written as a function of *u* and *u* can be written as a function of *x*) then

**Deﬁnition**: If a function F(*x*) is composed to two differentiable functions *g(x)* and *m*(*x*), so that F(*x*) = *g(m(x*)), then F(*x*) is differentiable and,

**Examples**

1. Compute , where diff((1 + 3 * x^2)(1/2), x, 1)

Let and , so and .

Then, by applying the chain rule, we have

2. Compute , where diff(sin(cos(x)), x, 1)

Let , so and .Then, by chain rule, we have

**Exercises**

Find the derivatives of the following function

- diff((x ^ 2+ 10) ^ (1/3), x, 1)
- diff(x ^ 2 * log(1 − 2 * x), x, 1)
- diff(x/sqrt(1 − 2 * x^2), x, 1)
- diff(tan(sqrt(2 − 3 * x)), x, 1)
- diff(%e ^ (−sin(1/x ^2)), x, 1)
- diff(log((1 − x ^ 2) ^ 2), x, 1)
- diff(%e ^ (%e ^ (2 ∗ x)), x, 1)
- diff(atan(3 * x ^2), x, 1)

**Higher derivatives**

**Deﬁnition**: Let *f* be a differentiable function, and let *f*′(x) be its derivative. The derivative of *f*′(x) (if it exists) is written *f*′′(x) and is called the second derivative of* f*. Similarly, the derivative of a second derivative, if it exists, is written *f*′′′(x) and is called the third derivative of *f*. These repeated derivatives are called higher-order derivatives.

**Examples**

1. Find the third derivative of with respect to x.

diff(x ^ 5 + 6 * x ^ 3 + 4 * x + 3, x, 3)

2. Find the third derivative of with respect to x.

diff(12 * sin(2 * x) + log(x + 2) + 2 * x, x, 3)

**Exercises**

1. Find the third derivative of diff((100 * x + 1) ^ 2, x, 3)

2. Find the third derivative of diff(sin(x), x, 3)

3. Find the third derivative of diff(tan(x), x, 3)

4. Find the third and -th derivative of find it yourself

5. Find the third and -th derivative of find it yourself

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