Chain Rule Practice Problems

Find the derivative of following with respect to x:

f(x) = (x⁶ + 4)⁵

• f '(x) = 5x⁵(x⁴ + 4)⁴

• f '(x) = 6x⁵(x⁶ + 4)⁴

• f '(x) = 30x⁵(x⁶ + 4)⁴

• f '(x) = 30x⁶(x⁶ + 4)⁴

Find the derivative of following with respect to x:

f(x) = (x³ - 2x)²

• f '(x) = 6x⁵ - 12x³ + 8x

• f '(x) = 6x⁵ - 16x³ + 8x

• f '(x) = x⁶ - 4x⁴ + 4x²

• f '(x) = 6x⁵ - 16x³ - 8x

Find the derivative of following with respect to x:

y = (2x + 1)¹⁰

• y' = 10(2x + 1)⁹

• y' = 20(2x - 1)⁹

• y' = 20(2x + 1)¹⁰

• y' = 20(2x + 1)⁹

Find the derivative of the following with respect to x:

y = sin(2x²)

• y' = 4sin(2x²)

• y' = 4xcos(2x²)

• y' = 2xcos(2x²)

• y' = xcos(4x²)

What is the "outer" part for this composite function:
 $2\left(x^3-1\right)^2$  

•  $2x^2$  

•  $x^3-1$  

• This is not a composite function.

What is the "inner" part of the composite function:
 $\left(\sin x\right)^2$  

•  $x^2$  

•  $\sin x$  

• This is not a composite function

What is the "outer" part of the composite function:
 $\sqrt{x^2+1}$  

•  $\sqrt{x}$  

•  $x^2+1$  

• This is not a composite function

Identify if the following would require the Chain Rule:
 $\frac{d}{dx}\left[f\left(x\right)\cdot g\left(x\right)\right]$  

• Yes

• No

Identify if the following would require the Chain Rule:
 $\frac{d}{dx}\left[h\left(t\left(x\right)\right)\right]$  

• Yes

• No

CHALLENGE:
Find the derivative of following with respect to x:
 $f\left(x\right)=\sqrt{\cos\left(x\right)}$  

• f '(x) =  $-\frac{\sin\left(x\right)}{2\cos^2\left(x\right)}$  

• f '(x) =  $\frac{2\sin\left(x\right)}{\sqrt{\cos\left(x\right)}}$  

• f '(x) =  $-\frac{\sin\left(x\right)}{2\sqrt{\cos\left(x\right)}}$  

• f '(x) =  $-\frac{\sin\left(x\right)}{\sqrt{\cos\left(x\right)}}$