Derivatives and gradient functions

Derivatives and gradient functions

general formula

Poll

understanding of previous lesson

• <50%

• >50%

Multiple Select

What is the average rate of change of  $f\left(x\right)=x^2+5x$  over the interval [1,5]?

•  $\frac{f\left(5\right)-f\left(1\right)}{5-1}$  

•  $\frac{50}{4}$  

• 11

• 4

Multiple Choice

 $f'\left(3\right)$  

Estimate 

•  $\frac{1}{3}$  

• 3

• 0

•  $-\frac{1}{3}$  

• -3

General Formula for Derivative

Prepare your note

General Formula for Derivative 

If  $f\left(x\right)=ax^n$ , then  $f'\left(x\right)=n\cdot ax^{n-1}$ 

 

Multiple Choice

Given $f\left(x\right)=2x^3-4x^2+7x-5$ , find  $f'\left(x\right)$ . 

•  $6x^2-8x+7-5$  

•  $6x^2-8x+7$  

•  $6x^3-8x^2+7x-5$  

•  $2x^2-4x+7$  

Multiple Select

Given $f\left(x\right)=x^2-6x+5$ , what is the gradient of tangent line at the point where  $x=1$   

•  $f'\left(1\right)$  

•  $f\left(1\right)$  

• 0

• - 4

Multiple Choice

Given  $f(x)=x^2−6x+5$ . 

Find the equation of tangent line at the point where x=1.
   

•  $y=-4x-4$  

•  $y=-4x+4$  

•  $y=4x-4$  

•  $y=4x+4$  

Multiple Choice

For a function g, we are given that  $g\left(7\right)=-3$   and  $g'\left(7\right)=-1$  
What's the equation of the tangent line to the graph of g at x=7?

•  $y-7=-1\left(x+3\right)$  

•  $y+3=-1\left(x-7\right)$  

•  $y-7=-3\left(x+1\right)$  

•  $y+1=-3\left(x-7\right)$  

The Chain rule

prepare your note

Multiple Choice

Find the derivative of  $y=\left(3x-2\right)^7$ .

•  $21\left(3x-2\right)^6$  

•  $\left(21x-14\right)^6$  

•  $21\left(3x-2\right)^7$  

•  $\left(21x-14\right)^7$  

•  $7\left(3x-2\right)^6$  

Multiple Choice

Find the derivative of  $\frac{1}{\left(3x^2+1\right)^5}$  

•  $-\frac{30x}{\left(3x^2+1\right)^6}$  

•  $-\frac{30x}{\left(3x^2+1\right)^5}$  

•  $\frac{30x}{\left(3x^2+1\right)^6}$  

•  $\frac{30x}{\left(3x^2+1\right)^4}$  

Multiple Choice

Find the derivative of  $\sqrt{\left(x-5\right)}$  

•  $\frac{1}{2\sqrt{\left(x-5\right)}}$  

•  $\frac{1}{2}\sqrt{\left(x-5\right)}$  

•  $-\frac{1}{2}\sqrt{\left(x-5\right)}$  

•  $-\frac{1}{2\sqrt{\left(x-5\right)}}$  

Multiple Choice

Find the derivative of  $\sqrt{\left(x-5\right)}$  

•  $\frac{1}{2\sqrt{\left(x-5\right)}}$  

•  $\frac{1}{2}\sqrt{\left(x-5\right)}$  

•  $-\frac{1}{2}\sqrt{\left(x-5\right)}$  

•  $-\frac{1}{2\sqrt{\left(x-5\right)}}$