Caluclus: Table of contents

Basic Properties of Derivatives

Continuity

Theorem (Continuity of differentiable functions) If a function \(f\) is differentiable at point \(a\) then it is continuous at \(a\).

Derivative of sum of functions

In this section we will prove that a sum of two differentiable functions is differentiable, and that a scalar multiple of a differentiable function is differentiable. We will also derive the properties for the derivative of a sum and the derivative of a scalar multiple of a function.

Theorem (Derivative of a scalar multiple) Assume that \(c\in\mathbb R\) and that \(f \colon\mathbb R\to\mathbb R\) is a differentiable function at point \(a\). Then \(g(x)=cf(x)\) is also differentiable at \(a\) and its derivative satisfies \(g^{\prime}(a)=cf^{\prime}(a)\).

Theorem (Derivative of a sum) If \(f\) and \(g\) are two differentiable functions at point \(a\), then so is \(h(x)=f(x)+g(x)\) and the derivative of \(h\) at \(a\) can be evaluated as: \[h^{\prime}(a)=f^{\prime}(a)+g^{\prime}(a).\]

From the previous two theorems we can conclude that \[(f-g)^{\prime}=\Big(f+(-1)\cdot g\Big)^{\prime}=f^{\prime}+(-1)\cdot g^{\prime}=f^{\prime}-g^{\prime}.\]

Product Rule

In this section we will derive the formula for the derivative of a product of two functions.

Theorem (Product rule) If \(f\) and \(g\) are two differentiable functions at \(a\in\mathbb R\) that so is \(h=f\cdot g\) and the following formula holds: \[h^{\prime}(a)=f^{\prime}(a)\cdot g(a)+f(a)\cdot g^{\prime}(a).\]

Example Find the derivative of the function \(f(x)=x^2\cdot \cos x\).

Theorem (Derivative of a reciprocal) If \(f\) is a function differentiable at \(a\) that satisfies \(f(a)\neq 0\), then the function \(g(x)=\frac1{f(x)}\) is also differentiable at \(a\) and satisfies \[g^{\prime}(a)=-\frac{f^{\prime}(a)}{f(a)^2}.\]

Using the previous two theorems we now establish the following result:

Theorem (Quotient rule) If \(f\) and \(g\) are function differentiable at \(a\) such that \(g(a)\neq 0\), then the function \(h(x)=\frac{f(x)}{g(x)}\) is differentiable at \(a\) and satisfies: \[h^{\prime}(a)=\frac{f^{\prime}(a)\cdot g(a)-f(a)\cdot g^{\prime}(a)}{g(a)^2}.\]

Practice problems

Problem 1. Determine the derivative of \(f(x)=x^3+3x^2-4x+18\).

Problem 2. Find the equation of the tangent line to the graph of \(f(x)=x+3\sqrt x\) at the point \((1,4)\).

Problem 3. Assume that \(f\), \(g\), and \(h\) are three differentiable functions. Derive the formula for the derivative of the function \(u(x)=f(x)g(x)h(x)\).

Problem 4. Let \(f(x)=\sin x\cdot \cos x+x^3\). Find \(f^{\prime}(x)\).

Problem 5. Let \(f(x)= \frac{x}{\cos x+x^3}\). Find \(f^{\prime}(x)\).