Question

How do I compute the derivatives of functions that are explicitly given with some formula?

Answer 1

To compute the derivatives of functions that are explicitly given by a formula, you can use a combination of basic differentiation rules and techniques. Here are some fundamental rules and examples to help you differentiate various types of functions:

1. Constant Rule: The derivative of a constant is always zero.
If \(f(x)=c\), where \(c\) is a constant, then \(f^{\prime}(x)=0\).

2. Power Rule: The derivative of a power function is found by multiplying the coefficient by the exponent and then reducing the exponent by one.
If \(f(x)=a x^n\), where \(a\) and \(n\) are constants, then \(f^{\prime}(x)=a n x^{n-1}\).
Example: \(f(x)=3 x^4, f^{\prime}(x)=12 x^3\)

3. Sum Rule: The derivative of the sum (or difference) of two functions is the sum (or difference) of their derivatives.
If \(f(x)=g(x)+h(x)\), then \(f^{\prime}(x)=g^{\prime}(x)+h^{\prime}(x)\)
Example: \(f(x)=x^3+2 x^2, f^{\prime}(x)=\left(3 x^2\right)+(4 x)\)

4. Product Rule: The derivative of a product of two functions is found by taking the derivative of the first function and multiplying it by the second function, and then adding the first function multiplied by the derivative of the second function.
If \(f(x)=g(x) h(x)\), then \(f^{\prime}(x)=g^{\prime}(x) h(x)+g(x) h^{\prime}(x)\)
Example: \(f(x)=x^2 \sin (x), f^{\prime}(x)=(2 x) \sin (x)+x^2 \cos (x)\).

5. Quotient Rule: The derivative of a quotient of two functions is found by taking the derivative of the numerator multiplied by the denominator, minus the numerator multiplied by the derivative of the denominator, all divided by the square of the denominator.
If \(f(x)=\frac{g(x)}{h(x)}\), then \(f^{\prime}(x)=\frac{g^{\prime}(x) h(x)-g(x) h^{\prime}(x)}{(h(x))^2}\).
Example: \(f(x)=\frac{x}{x^2+1}, f^{\prime}(x)=\frac{(1)\left(x^2+1\right)-x(2 x)}{\left(x^2+1\right)^2}\)

6. Chain Rule: The derivative of a composite function is found by taking the derivative of the outer function with respect to the inner function, multiplied by the derivative of the inner function.

If \(f(x)=g(h(x))\), then \(f^{\prime}(x)=g^{\prime}(h(x)) h^{\prime}(x)\)

Example: \(f(x)=\sin \left(x^2\right), f^{\prime}(x)=\left(\cos \left(x^2\right)\right)(2 x)\)
By applying these rules and techniques, you can compute the derivatives of various functions that are explicitly given by a formula. In some cases, you may need to simplify the resulting expression to obtain the final form of the derivative.