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Calculus- Simple Differentiation

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Question 1


Use the result that the derivative of $y=mx+c$ is $\frac{\mathrm{d}y}{\mathrm{d}x}=m$ to differentiate the following:

a) $y=2x+1$  a s v d
b) $y=3x+2$  a s v d
c) $y-x+4$  a s v d
d) $y=-5x+4$  a s v d
e) $y=7x-2$  a s v d
f) $y=-20x-18$  a s v d
g) $y=4x$  a s v d
h) $y=x$  a s v d
i) $y=0.5x+7$  a s v d
j) $y=\tfrac{1}{3}x+15$  a s v d
k) $y=-\tfrac{5}{6}-\tfrac{6}{25}$  a s v d
l) $y=\sqrt2 x$  a s v d
m) $y=\pi x + \sqrt2$  a s v d


Question 2


Simplify into the form $y=mx+c$ before differentiating.

a) $y=x\times 6 -9$  a s v d
b) $y=x \times \sqrt3 -17$  a s v d
c) $y=x+x$  a s v d
d) $y=5x+2-x$  a s v d
e) $y-x=2$  a s v d
f) $y-15=20+x \times 5 - 7x$  a s v d
g) $y=\tfrac{1}{7} x + x \times 5 +17 \times 2$  a s v d


Question 3


Use the result that the derivative of $y=x^n$ is $\frac{\mathrm{d}y}{\mathrm{d}x}=nx^{n-1}$ to differentiate the following:

a) $y=x^3$  a s v d
b) $y=x^4$  a s v d
c) $y=x^2$  a s v d
d) $y=x^1$  a s v d
e) $y=x$  a s v d
f) $y=x^{20}$  a s v d
g) $y=x^{-2}$  a s v d
h) $y=x^{-5}$  a s v d
i) $y=x^{a}$  a s v d
j) $y=x^{\frac32}$  a s v d
k) $y=x^{\frac12}$  a s v d
l) $y=x^{b+5}$  a s v d


Question 4


Simplify into the form $y=x^n$ before differentiating.

a) $y=x\times x$  a s v d
b) $y=x^2\times x^3$  a s v d
c) $y=\frac1x$  a s v d
d) $y=\frac{1}{x^2}$  a s v d
e) $y=x^6 \times \frac{1}{x^2}$  a s v d
f) $y=\frac{1}{x^2 \times x^5}$  a s v d
g) $y=\sqrt x$  a s v d
h) $y=\sqrt[3]{x}$  a s v d
i) $y=\frac{1}{\sqrt[4]{x}}$  a s v d


Question 5


Use the result that the derivative of $y=ax^n$ is $\frac{\mathrm{d}y}{\mathrm{d}x}=anx^{n-1}$ to differentiate the following:

a) $y=2x^3$  a s v d
b) $y=5x^7$  a s v d
c) $y=-3x^5$  a s v d
d) $y=-2x^{-3}$  a s v d
e) $y=4x^{\frac12}$  a s v d
f) $y=15x^{\frac15}$  a s v d
g) $y=\frac12x^{\frac53}$  a s v d
h) $y=rx^{s}$  a s v d


Question 6


Use the result that the derivative of $y=f(x)+g(x)$ is $\frac{\mathrm{d}y}{\mathrm{d}x}=f'(x)+g'(x)$ to differentiate the following:

a) $y=x^2+x^3$  a s v d
b) $y=x^6+x^{20}$  a s v d
c) $y=2x^3-x^5$  a s v d
d) $y=5x^4-2x^{-4}$  a s v d
e) $y=x^{\frac73}-\tfrac12x^6$  a s v d
f) $y=x^2+x^3+x^4$  a s v d
g) $y=2x^2-5x+3$  a s v d
h) $y=16x+\tfrac45x^{25}-4x^{-\frac29}+52$  a s v d
i) $y=mx^n+px^q$  a s v d


Question 7


Simplify before differentiating.

a) $y=x+2x+3x^2-9$  a s v d
b) $y=\frac{x^2}{x}+22$  a s v d
c) $y=\frac{2x}{x^4}-x$  a s v d
d) $y=\frac{5}{2x^3}$  a s v d
e) $y=\frac{4}{x^2}\times\frac{1}{2x}$  a s v d


Question 8


Answer the following worded questions:

a) Find the derivative of $y=6x^3$  a s v d

b) What is the gradient of $y=2x$?  a s v d

c) Find the gradient of $y=-6x+4$  a s v d

d) A curve is defined as $f(x)=x^2+3$. Find $f'(x)$  a s v d

e) Find the derivate of $g(a)=3-4a^4$  a s v d

f) Find $f'(1)$ for $f(x)=3x^7+18$  a s v d

g) Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ for $y=-\frac{1}{\sqrt x}$  a s v d

h) What is the gradient of $y=5x^2+2x$ when $x=2$?  a s v d

i) Find the gradient of $y=6\sqrt x$ at the point $(4,12)$  a s v d