# 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 $$f^{\prime}(x)$$.