Calculus 1

Differentiate  $y=e^{7x}$  .

•  $\frac{\text{d}y}{\text{d}x}=e^{7x}$  

•  $\frac{dy}{\text{d}x}=7e^x$  

•  $\frac{\text{d}y}{\text{d}x}=\frac{1}{7}e^x$  

•  $\frac{\text{d}y}{\text{d}x}=7e^{7x}$  

Differentiate  $f\left(x\right)=x^3\left(x-7\right)\left(2x+1\right)$  

•  $f'\left(x\right)=3x^2\left(2x^2-11x-21\right)-x^3\left(4x-11\right)$  

•  $f'\left(x\right)=3x^2\left(x-7\right)\left(2x+3\right)+x^3\left(2x+3\right)+x^3\left(x-7\right)$  

•  $f'\left(x\right)=10x^4-44x^3-63x^2$  

•  $f'\left(x\right)=3x^2\left(x-7\right)\left(2x+3\right)+x^3\left(4x-7\right)$  

 $\lim_{x\rightarrow7}\ \frac{x-7}{\sqrt{x-3}-2}$  

• 0

• 4

• Does not exist.

•  $\frac{4}{7}$  

Differentiate  $y=4^{2x-3}$  

•  $y'=2\ln4\cdot4^{2x-3}$  

•  $y'=\left(2x-3\right)\ln4\cdot4^{2x-3}$  

•  $y'=\ln4\cdot4^{2x-3}$  

•  $y'=\ln8\cdot4^{2x-3}$  

Differentiate  $y=e^{x^2\sin x^3}$  

•  $\frac{\text{d}y}{\text{d}x}=e^{x^2\sin x^3}\cdot\left(2x\cos x^3\right)$  

•  $\frac{\text{d}y}{\text{d}x}=e^{x^2\sin x^3}$  

•  $\frac{\text{d}y}{\text{d}x}=e^x\cdot\left(5x^4\cos x^3\right)$  

•  $\frac{\text{d}y}{\text{d}x}=e^{x^2\sin x^3}\cdot\left(2x\sin x^3+3x^4\cos x^3\right)$  

 $\lim_{x\rightarrow1}\ \frac{x^{\frac{1}{3}}-1}{x-1}$  

•  $\frac{1}{3}$  

•  $\frac{1}{4}$  

•  $3$  

• Does not exist.

Determine the initial acceleration of a particle whose position is given by  $s\left(t\right)=-e^{5t}+\sin t$  , where  $s$  is the distance traveled in metres,  $t$  seconds after it starts moving.

•  $-e^5\ \frac{m}{s^2}$  

•  $-25\ \frac{m}{s^2}$  

•  $-5e^5+1\ \frac{m}{s^2}$  

•  $0\ \frac{m}{s^2}$  

 Differentiate $y=\sqrt{e^x-\cos^2x+\tan5x}$ 

•  $\frac{\text{d}y}{\text{d}x}=\frac{e^x+2\cos x\sin x+5\sec^2\left(5x\right)}{2\sqrt{e^x-\cos^2x+\tan5x}}$  

•  $\frac{\text{d}y}{\text{d}x}=\frac{1}{2}e^{\frac{x}{2}}+\sin x+\frac{5}{2}\sec^2\left(5x\right)$  

•  $\frac{\text{d}y}{\text{d}x}=\frac{1}{2\sqrt{e^x}}+\sin x+\frac{5}{2}\tan\left(5x\right)\sec^2\left(5x\right)$  

•  $\frac{\text{d}y}{\text{d}x}=\frac{\left(e^x+2\cos x\sin x+5\sec^2\left(5x\right)\right)^{-\frac{1}{2}}}{2}$  

Determine the slope of the tangent line to the curve of $y=\cos\left(\frac{t}{\left(2\right)}\right)$  when  $t=\frac{\pi}{2}$  .

•  $-1$  

•  $-\frac{\sqrt{2}}{4}$  

•  $-\frac{\sqrt{2}}{2}$  

•  $0$  

 Differentiate $f\left(x\right)=\frac{x^2-5x+6}{x-3}$ .

•  $f'\left(x\right)=\frac{\left(2x-5\right)\left(x-3\right)+\left(x^2-5x+6\right)}{\left(x-3\right)^2}$  

•  $f'\left(x\right)=2x-5$  

•  $f'\left(x\right)=1;\ x\ne3$  

•  $f'\left(x\right)=x+2;\ x\ne3$