Differentiation

Finding the derivative

Derivative of a function tells us how steep is this function at a given point.
Derivative of the function \(f(x)=x^n\) is:

\(f'(x)=n\,x^{n-1}\)

Derivative can be written as \(f'(x)\) or \(y'\) or \(\frac{\mathrm{d}y}{\mathrm{d}x}\). The procedure of finding a derivative is called differentiation.

Find derivatives of the following functions:

(a) \(f(x)=x^3-3x^2+4x+1\)

(b) \(f(x)=x^4+2x^3-5x\)

Solutions: (a) \(f'(x)=3x^2-6x+4\); (b) \(f'(x)=4x^3+6x^2-5\); (c) \(f'(x)=-4x+5\)

Differentiate the following functions:

(a) \(f(x)=\frac{1}{2}x^2+3x-7\)

(b) \(f(x)=x^2(x^2+x-1)\)

Solutions: (a) \(f'(x)=x+3\); (b) \(f'(x)=4x^3+3x^2-2x\); (c) \(f'(x)=18x+12\)

Find \(y'\) of the following functions:

(a) \(f(x)=x^2+\frac{\textstyle 1}{\textstyle x}\)

(b) \(f(x)=\frac{\textstyle 2}{\textstyle x^2}+\frac{\textstyle 1}{\textstyle x^3}\)

Solutions: (a) \(f'(x)=2x-\frac{1}{x^2}\); (b) \(f'(x)=-\frac{4}{x^3}-\frac{3}{x^4}\); (c) \(f'(x)=1-\frac{1}{x^2}\)

Find \(\frac{\textstyle \mathrm{d}y}{\textstyle \mathrm{d}x}\) for the following functions:

(a) \(y=\sqrt{x}\)

(b) \(y=\sqrt[\scriptstyle 3]{x}+\sqrt[\scriptstyle 4]{x}\)

Solutions: (a) \(\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{2\sqrt{x}}\); (b) \(\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{3\,\sqrt[3]{x^2}}+\frac{1}{4\,\sqrt[4]{x^3}}\); (c) \(\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{3}{\sqrt{x^5}}\)

Derivatives of functions \(e^x\) and \(\ln x\) are:

\((e^x)'=e^x\)

\((\ln x)'=\frac{1}{x}\)

Find derivatives of the following functions:

(a) \(f(x)=2x+e^x\)

(b) \(f(x)=5e^x+x^2\)

(d) \(f(x)=3e^x+2\ln x\)

Solutions: (a) \(f'(x)=2+e^x\); (b) \(f'(x)=5e^x+2x\); (c) \(f'(x)=3x^2+\frac{1}{x}\); (d) \(f'(x)=3e^x+\frac{2}{x}\)

Tangents and normals

Derivative of a function is equal to the gradient of the tangent at a given point:

\(m=f'(x_1)\)

Equation of the tangent can be found using the form:

\(y=m\,x+c\) or \(y-y_1=m\,(x-x_1)\)

Consider the function \(f(x)=x^3-x+1\) and point \(P(1,1)\) which lies on the graph of this function.

(a) Find the derivative \(f'(x)\).

(b) Calculate the gradient of the tangent.

Solutions: (a) \(f'(x)=3x^2-1\); (b) \(m=f'(1)=2\); (c) \(y=2x-1\)

Find the equation of the tangent to the curve \(y=x^3-3x+3\) at given points:

(a) \(P(0,3)\)

(b) \(P(2,y)\)

Solutions: (a) \(y=-3x+3\); (b) \(y=9x-13\); (c) \(y=1\)

Find the equation of the tangent to the graph of the function \(f(x)=\frac{1}{3}x^2+x-2\) at the point \(P(3,y)\).
Solution: Tangent: \(y=3x-5\)

Find the equation of the tangent to the graph of the function \(y=x^3+2x^2\) at the point \(x=1\).
Solution: Tangent: \(y=7x-4\)

Find the equation of the tangent to the graph of the function \(y=x-x^{-1}+1\) at the point \(x=-1\).
Solution: Tangent: \(y=2x+3\)

Find the equation of the tangent to the graph of the function \(y=\sqrt{x}\) at the point \(x=9\).
Solution: Tangent: \(y=\frac{1}{6}x+\frac{3}{2}\)

The function has the equation: \(f(x)=\frac{\textstyle x^2+x+1}{\textstyle 3}\).

(a) Differentiate the function \(f\).

(b) Write down the equation of the tangent at \(x=4\).

Solutions: (a) \(f'(x)=\frac{2x+1}{3}\); (b) tangent: \(y=3x-5\); (c) they have the same gradient: \(m_1=m_2=3\)

derivative function on your GDC

Find the equation of the tangent to the graph of the function \(f(x)=\sqrt{2x+3}\) at \(x=-1\).
Solution: Tangent: \(y=x+2\)

Write the equation of the tangent to the graph of the function \(f(x)=(x+2)\sqrt{x}\) at \(x=4\).
Solution: Tangent: \(y=\frac{7}{2}x-2\)

Find the equation of the tangent to the graph of the function \({\displaystyle f(x)=\frac{2}{x-1}}\) at \(x=2\).
Solution: Tangent: \(y=-2x+6\)

Write the equation of the tangent to the graph of the function \({\displaystyle f(x)=\frac{2x-3}{3x+1}}\) at the point where the graph intercepts the ordinate axis.
Solution: Tangent: \(y=11x-3\)

Write the equation of the tangent to the graph of the function \({\displaystyle f(x)=\frac{e^x}{x-1}}\) at \(x=0\).
Solution: Tangent: \(y=-2x-1\)

Find the equation of the tangent to the graph of the function \({\displaystyle y=\frac{1+e^x}{1-x}}\) at \(x=0\).
Solution: Tangent: \(y=3x+2\)

Find the equation of the tangent to the curve \({\displaystyle y=\frac{x-3}{\sqrt{x}}}\)

(a) at \(x=1\),

(b) at \(x=9\).

Solutions: (a) \(y=2x-4\); (b) \(y=\frac{2}{9}x\)

Write the equation of the tangent to the graph of the function \(f(x)=\frac{\textstyle x\ln x}{\textstyle 2}\) at the point \(P(1,y)\).
Solution: Tangent: \(y=\frac{1}{2}x-\frac{1}{2}\)

Write the equation of the tangent to the graph of function \(f(x)=\ln(x^2-3)\) at point \(P(2,y)\).
Solution: Tangent: \(y=4x-8\)

Normal is perpendicular to the graph.
The gradient of the normal is:

\({\displaystyle m_n=-\,\frac{1}{m_t}}\)

Consider the function \(f(x)=x^3-x+1\).

(a) Draw the graph of this function.

(b) Find the equation of the tangent at \(x=1\).

Solutions: (b) Tangent \(y=2x-1\); (c) normal: \(y=-\frac{1}{2}x+\frac{3}{2}\)

Write the equation of the normal to the curve \({\displaystyle y=\frac{2x}{2-x^2}}\) at the point \(P(2,y)\).
Solution: Normal: \(y=-\frac{1}{3}x-\frac{4}{3}\)

Find the equation of the normal to the graph of the function \(f(x)=\ln(x-3)\) at \(x=4\).
Solution: \(y=-x+4\)

Find the equation of the normal to the graph of the function \(f(x)=x\sqrt{x+4}\) at \(x=-3\).
Solution: \(y=2x+3\)

Find the equation of the normal to the graph of the function \(y=(x-2)\sqrt{x+2}\) at \(x\)-axis intercept with positive abscissa.
Solution: \(y=-\frac{1}{2}x+1\)

Write the equation of the normal to the curve \({\displaystyle y=\frac{1}{~\sqrt[\scriptstyle3]{x-9}~}}\) at the point \(P_0(8,y_0)\).
Solution: Normal: \(y=3x-25\)

Function \(f\) has the equation \({\displaystyle f(x)=1+\frac{8}{x+2}}\)

(a) Draw the graph of this function.

(b) Write down the equation of the normal at point \(A(2,y)\).

Solutions: (b) \(y=2x-1\); (c) \(B(-3,-7)\)

Consider the function \({\displaystyle y=\frac{x}{x+2}}\)

(a) Draw the graph of this function.

(b) Write down the equation of the normal at \(x=0\).

(d) Find the intersection point of these two normals.

Solutions: (b) \(y=-2x\); (c) \(y=-\frac{1}{2}x+\frac{3}{2}\); (d) \(P(-1,2)\)

The function has the equation: \(f(x)=\sqrt{2x+7}\).

(a) Write down the equation of the normal at \(x=1\).

This normal and both coordinate axes form a triangle.

(b) Write down the coordinates of the vertices of this triangle.

Solutions: (a) normal: \(y=-3x+6\); (b) vertices: \(P_1(0,0),~ P_2(2,0),~ P_3(0,6)\); (c) area: \(A=6\)

Finding points with given gradient

Consider the function \(f(x)=x^2-3x-2\).

(a) Find the derivative \(f'(x)\).

(b) Hence, find the point on the graph where the gradient is 1.

Solutions: (a) \(f'(x)=2x-3\); (b) \(P(2,-4)\)

The function has the equation \(f(x)=x^3-7x+1\). Find the points on the graph where the gradient of this function is 5. Write down the coordinates of these points.
Solutions: At points \(P_1(2,-5)\) and \(P_2(-2,7)\).

The function has the equation \(f(x)=x^3-6x^2+9x+1\). Find the points on the graph where the gradient is 0. Write down the coordinates of these points.
Solutions: \(P_1(1,5),~ P_2(3,1)\)

The function has the equation \(f(x)=\frac{1}{3}x^3-\frac{1}{2}x^2+1\). Find the points on the graph where the tangent is parallel to the line \(y=2x\). Write down the coordinates of these points.
Solutions: \(P_1(-1,\frac{1}{6}),~ P_2(2,\frac{5}{3})\)

The function has the equation \(f(x)=x^3-3x^2\). Find the point on the graph where the tangent is parallel to the line \(y=-3x\). Write down the coordinates of this point.
Solution: \(P(1,-2)\)