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) $y=2x+1$
b) $y=3x+2$
c) $y-x+4$
d) $y=-5x+4$
e) $y=7x-2$
f) $y=-20x-18$
g) $y=4x$
h) $y=x$
i) $y=0.5x+7$
j) $y=\tfrac{1}{3}x+15$
k) $y=-\tfrac{5}{6}-\tfrac{6}{25}$
l) $y=\sqrt2 x$
m) $y=\pi x + \sqrt2$
Question 2
Simplify into the form $y=mx+c$ before differentiating.
a) $y=x\times 6 -9$
b) $y=x \times \sqrt3 -17$
c) $y=x+x$
d) $y=5x+2-x$
e) $y-x=2$
f) $y-15=20+x \times 5 - 7x$
g) $y=\tfrac{1}{7} x + x \times 5 +17 \times 2$
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) $y=x^3$
b) $y=x^4$
c) $y=x^2$
d) $y=x^1$
e) $y=x$
f) $y=x^{20}$
g) $y=x^{-2}$
h) $y=x^{-5}$
i) $y=x^{a}$
j) $y=x^{\frac32}$
k) $y=x^{\frac12}$
l) $y=x^{b+5}$
Question 4
Simplify into the form $y=x^n$ before differentiating.
a) $y=x\times x$
b) $y=x^2\times x^3$
c) $y=\frac1x$
d) $y=\frac{1}{x^2}$
e) $y=x^6 \times \frac{1}{x^2}$
f) $y=\frac{1}{x^2 \times x^5}$
g) $y=\sqrt x$
h) $y=\sqrt[3]{x}$
i) $y=\frac{1}{\sqrt[4]{x}}$
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) $y=2x^3$
b) $y=5x^7$
c) $y=-3x^5$
d) $y=-2x^{-3}$
e) $y=4x^{\frac12}$
f) $y=15x^{\frac15}$
g) $y=\frac12x^{\frac53}$
h) $y=rx^{s}$
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) $y=x^2+x^3$
b) $y=x^6+x^{20}$
c) $y=2x^3-x^5$
d) $y=5x^4-2x^{-4}$
e) $y=x^{\frac73}-\tfrac12x^6$
f) $y=x^2+x^3+x^4$
g) $y=2x^2-5x+3$
h) $y=16x+\tfrac45x^{25}-4x^{-\frac29}+52$
i) $y=mx^n+px^q$
Question 7
Simplify before differentiating.
a) $y=x+2x+3x^2-9$
b) $y=\frac{x^2}{x}+22$
c) $y=\frac{2x}{x^4}-x$
d) $y=\frac{5}{2x^3}$
e) $y=\frac{4}{x^2}\times\frac{1}{2x}$
Question 8
Answer the following worded questions:
a) Find the derivative of $y=6x^3$
b) What is the gradient of $y=2x$?
c) Find the gradient of $y=-6x+4$
d) A curve is defined as $f(x)=x^2+3$. Find $f'(x)$
e) Find the derivate of $g(a)=3-4a^4$
f) Find $f'(1)$ for $f(x)=3x^7+18$
g) Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ for $y=-\frac{1}{\sqrt x}$
h) What is the gradient of $y=5x^2+2x$ when $x=2$?
i) Find the gradient of $y=6\sqrt x$ at the point $(4,12)$
Answers
Question 1
a) $\frac{\mathrm{d}y}{\mathrm{d}x}=2$
b) $\frac{\mathrm{d}y}{\mathrm{d}x}=3$
c) $\frac{\mathrm{d}y}{\mathrm{d}x}=1$
d) $\frac{\mathrm{d}y}{\mathrm{d}x}=-5$
e) $\frac{\mathrm{d}y}{\mathrm{d}x}=7$
f) $\frac{\mathrm{d}y}{\mathrm{d}x}=-20$
g) $\frac{\mathrm{d}y}{\mathrm{d}x}=4$
h) $\frac{\mathrm{d}y}{\mathrm{d}x}=1$
i) $\frac{\mathrm{d}y}{\mathrm{d}x}=0.5$
j) $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{3}$
k) $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{5}{6}$
l) $\frac{\mathrm{d}y}{\mathrm{d}x}=\sqrt2$
m) $\frac{\mathrm{d}y}{\mathrm{d}x}=\pi$
Question 2
a) $\frac{\mathrm{d}y}{\mathrm{d}x}=6$
b) $\frac{\mathrm{d}y}{\mathrm{d}x}=\sqrt3$
c) $\frac{\mathrm{d}y}{\mathrm{d}x}=2$
d) $\frac{\mathrm{d}y}{\mathrm{d}x}=4$
e) $\frac{\mathrm{d}y}{\mathrm{d}x}=1$
f) $\frac{\mathrm{d}y}{\mathrm{d}x}=-2$
g) $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{36}{7}$ or $\frac{\mathrm{d}y}{\mathrm{d}x}=5\frac17$
Question 3
a) $\frac{\mathrm{d}y}{\mathrm{d}x}=3x^2$
b) $\frac{\mathrm{d}y}{\mathrm{d}x}=4x^3$
c) $\frac{\mathrm{d}y}{\mathrm{d}x}=2x$
d) $\frac{\mathrm{d}y}{\mathrm{d}x}=1$
e) $\frac{\mathrm{d}y}{\mathrm{d}x}=1$
f) $\frac{\mathrm{d}y}{\mathrm{d}x}=20x^{19}$
g) $\frac{\mathrm{d}y}{\mathrm{d}x}=-2x^{-3}$
h) $\frac{\mathrm{d}y}{\mathrm{d}x}=-5x^{-6}$
i) $\frac{\mathrm{d}y}{\mathrm{d}x}=ax^{a-1}$
j) $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac32 x^{\frac12}$
k) $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac12 x^{-\frac12}$
l) $\frac{\mathrm{d}y}{\mathrm{d}x}=(b+5)x^{b+4}$
Question 4
a) $\frac{\mathrm{d}y}{\mathrm{d}x}=2x$
b) $\frac{\mathrm{d}y}{\mathrm{d}x}=5x^4$
c) $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{1}{x^2}$
d) $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{2}{x^3}$
e) $\frac{\mathrm{d}y}{\mathrm{d}x}=4x^3$
f) $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{7}{x^8}$
g) $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac12 x^{-\frac12}$
h) $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac13 x^{-\frac23}$
i) $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{1}{4\sqrt[4]{x^5}}$
Question 5
a) $\frac{\mathrm{d}y}{\mathrm{d}x}=6x^2$
b) $\frac{\mathrm{d}y}{\mathrm{d}x}=35x^6$
c) $\frac{\mathrm{d}y}{\mathrm{d}x}=-15x^4$
d) $\frac{\mathrm{d}y}{\mathrm{d}x}=6x^{-4}$
e) $\frac{\mathrm{d}y}{\mathrm{d}x}=2x^{-\frac12}$
f) $\frac{\mathrm{d}y}{\mathrm{d}x}=3x^{-\frac45}$
g) $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac56 x^{\frac23}$
h) $\frac{\mathrm{d}y}{\mathrm{d}x}=rsx^{s-1}$
Question 6
a) $\frac{\mathrm{d}y}{\mathrm{d}x}=2x+3x^2$
b) $\frac{\mathrm{d}y}{\mathrm{d}x}=6x^5+20x^{19}$
c) $\frac{\mathrm{d}y}{\mathrm{d}x}=6x^2-5x^4$
d) $\frac{\mathrm{d}y}{\mathrm{d}x}=20x^3+8x^-5$
e) $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac73 x^{\frac43}-3x^5$
f) $\frac{\mathrm{d}y}{\mathrm{d}x}=2x+3x^2+4x^3$
g) $\frac{\mathrm{d}y}{\mathrm{d}x}=4x-5$
h) $\frac{\mathrm{d}y}{\mathrm{d}x}=16+20x^{24}+\frac89x^{-\frac{11}{9}}$
i) $\frac{\mathrm{d}y}{\mathrm{d}x}=mn^{n-1}+pqx^{q-1}$
Question 7
a) $\frac{\mathrm{d}y}{\mathrm{d}x}=3+6x$
b) $\frac{\mathrm{d}y}{\mathrm{d}x}=1$
c) $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{8}{x^5}-1$
d) $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{15}{2x^4}$
e) $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{6}{x^{-4}}$
Question 8
a) The derivative is $\frac{\mathrm{d}y}{\mathrm{d}x}=18x^2$
b) $2$
c) $-6$
d) $f'(x)=2x$
e) $g'(a)=-16a^3$
f) $f'(1)=21$
g) $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{2\sqrt{x^3}}$
h) The gradient it $22$
i) The gradeint is $\frac34$
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) $y=2x+1$
b) $y=3x+2$
c) $y-x+4$
d) $y=-5x+4$
e) $y=7x-2$
f) $y=-20x-18$
g) $y=4x$
h) $y=x$
i) $y=0.5x+7$
j) $y=\tfrac{1}{3}x+15$
k) $y=-\tfrac{5}{6}-\tfrac{6}{25}$
l) $y=\sqrt2 x$
m) $y=\pi x + \sqrt2$
Question 2
Simplify into the form $y=mx+c$ before differentiating.
a) $y=x\times 6 -9$
b) $y=x \times \sqrt3 -17$
c) $y=x+x$
d) $y=5x+2-x$
e) $y-x=2$
f) $y-15=20+x \times 5 - 7x$
g) $y=\tfrac{1}{7} x + x \times 5 +17 \times 2$
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) $y=x^3$
b) $y=x^4$
c) $y=x^2$
d) $y=x^1$
e) $y=x$
f) $y=x^{20}$
g) $y=x^{-2}$
h) $y=x^{-5}$
i) $y=x^{a}$
j) $y=x^{\frac32}$
k) $y=x^{\frac12}$
l) $y=x^{b+5}$
Question 4
Simplify into the form $y=x^n$ before differentiating.
a) $y=x\times x$
b) $y=x^2\times x^3$
c) $y=\frac1x$
d) $y=\frac{1}{x^2}$
e) $y=x^6 \times \frac{1}{x^2}$
f) $y=\frac{1}{x^2 \times x^5}$
g) $y=\sqrt x$
h) $y=\sqrt[3]{x}$
i) $y=\frac{1}{\sqrt[4]{x}}$
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) $y=2x^3$
b) $y=5x^7$
c) $y=-3x^5$
d) $y=-2x^{-3}$
e) $y=4x^{\frac12}$
f) $y=15x^{\frac15}$
g) $y=\frac12x^{\frac53}$
h) $y=rx^{s}$
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) $y=x^2+x^3$
b) $y=x^6+x^{20}$
c) $y=2x^3-x^5$
d) $y=5x^4-2x^{-4}$
e) $y=x^{\frac73}-\tfrac12x^6$
f) $y=x^2+x^3+x^4$
g) $y=2x^2-5x+3$
h) $y=16x+\tfrac45x^{25}-4x^{-\frac29}+52$
i) $y=mx^n+px^q$
Question 7
Simplify before differentiating.
a) $y=x+2x+3x^2-9$
b) $y=\frac{x^2}{x}+22$
c) $y=\frac{2x}{x^4}-x$
d) $y=\frac{5}{2x^3}$
e) $y=\frac{4}{x^2}\times\frac{1}{2x}$
Question 8
Answer the following worded questions:
a) Find the derivative of $y=6x^3$
b) What is the gradient of $y=2x$?
c) Find the gradient of $y=-6x+4$
d) A curve is defined as $f(x)=x^2+3$. Find $f'(x)$
e) Find the derivate of $g(a)=3-4a^4$
f) Find $f'(1)$ for $f(x)=3x^7+18$
g) Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ for $y=-\frac{1}{\sqrt x}$
h) What is the gradient of $y=5x^2+2x$ when $x=2$?
i) Find the gradient of $y=6\sqrt x$ at the point $(4,12)$
Answers
Question 1
a) $\frac{\mathrm{d}y}{\mathrm{d}x}=2$
b) $\frac{\mathrm{d}y}{\mathrm{d}x}=3$
c) $\frac{\mathrm{d}y}{\mathrm{d}x}=1$
d) $\frac{\mathrm{d}y}{\mathrm{d}x}=-5$
e) $\frac{\mathrm{d}y}{\mathrm{d}x}=7$
f) $\frac{\mathrm{d}y}{\mathrm{d}x}=-20$
g) $\frac{\mathrm{d}y}{\mathrm{d}x}=4$
h) $\frac{\mathrm{d}y}{\mathrm{d}x}=1$
i) $\frac{\mathrm{d}y}{\mathrm{d}x}=0.5$
j) $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{3}$
k) $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{5}{6}$
l) $\frac{\mathrm{d}y}{\mathrm{d}x}=\sqrt2$
m) $\frac{\mathrm{d}y}{\mathrm{d}x}=\pi$
Question 2
a) $\frac{\mathrm{d}y}{\mathrm{d}x}=6$
b) $\frac{\mathrm{d}y}{\mathrm{d}x}=\sqrt3$
c) $\frac{\mathrm{d}y}{\mathrm{d}x}=2$
d) $\frac{\mathrm{d}y}{\mathrm{d}x}=4$
e) $\frac{\mathrm{d}y}{\mathrm{d}x}=1$
f) $\frac{\mathrm{d}y}{\mathrm{d}x}=-2$
g) $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{36}{7}$ or $\frac{\mathrm{d}y}{\mathrm{d}x}=5\frac17$
Question 3
a) $\frac{\mathrm{d}y}{\mathrm{d}x}=3x^2$
b) $\frac{\mathrm{d}y}{\mathrm{d}x}=4x^3$
c) $\frac{\mathrm{d}y}{\mathrm{d}x}=2x$
d) $\frac{\mathrm{d}y}{\mathrm{d}x}=1$
e) $\frac{\mathrm{d}y}{\mathrm{d}x}=1$
f) $\frac{\mathrm{d}y}{\mathrm{d}x}=20x^{19}$
g) $\frac{\mathrm{d}y}{\mathrm{d}x}=-2x^{-3}$
h) $\frac{\mathrm{d}y}{\mathrm{d}x}=-5x^{-6}$
i) $\frac{\mathrm{d}y}{\mathrm{d}x}=ax^{a-1}$
j) $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac32 x^{\frac12}$
k) $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac12 x^{-\frac12}$
l) $\frac{\mathrm{d}y}{\mathrm{d}x}=(b+5)x^{b+4}$
Question 4
a) $\frac{\mathrm{d}y}{\mathrm{d}x}=2x$
b) $\frac{\mathrm{d}y}{\mathrm{d}x}=5x^4$
c) $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{1}{x^2}$
d) $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{2}{x^3}$
e) $\frac{\mathrm{d}y}{\mathrm{d}x}=4x^3$
f) $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{7}{x^8}$
g) $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac12 x^{-\frac12}$
h) $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac13 x^{-\frac23}$
i) $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{1}{4\sqrt[4]{x^5}}$
Question 5
a) $\frac{\mathrm{d}y}{\mathrm{d}x}=6x^2$
b) $\frac{\mathrm{d}y}{\mathrm{d}x}=35x^6$
c) $\frac{\mathrm{d}y}{\mathrm{d}x}=-15x^4$
d) $\frac{\mathrm{d}y}{\mathrm{d}x}=6x^{-4}$
e) $\frac{\mathrm{d}y}{\mathrm{d}x}=2x^{-\frac12}$
f) $\frac{\mathrm{d}y}{\mathrm{d}x}=3x^{-\frac45}$
g) $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac56 x^{\frac23}$
h) $\frac{\mathrm{d}y}{\mathrm{d}x}=rsx^{s-1}$
Question 6
a) $\frac{\mathrm{d}y}{\mathrm{d}x}=2x+3x^2$
b) $\frac{\mathrm{d}y}{\mathrm{d}x}=6x^5+20x^{19}$
c) $\frac{\mathrm{d}y}{\mathrm{d}x}=6x^2-5x^4$
d) $\frac{\mathrm{d}y}{\mathrm{d}x}=20x^3+8x^-5$
e) $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac73 x^{\frac43}-3x^5$
f) $\frac{\mathrm{d}y}{\mathrm{d}x}=2x+3x^2+4x^3$
g) $\frac{\mathrm{d}y}{\mathrm{d}x}=4x-5$
h) $\frac{\mathrm{d}y}{\mathrm{d}x}=16+20x^{24}+\frac89x^{-\frac{11}{9}}$
i) $\frac{\mathrm{d}y}{\mathrm{d}x}=mn^{n-1}+pqx^{q-1}$
Question 7
a) $\frac{\mathrm{d}y}{\mathrm{d}x}=3+6x$
b) $\frac{\mathrm{d}y}{\mathrm{d}x}=1$
c) $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{8}{x^5}-1$
d) $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{15}{2x^4}$
e) $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{6}{x^{-4}}$
Question 8
a) The derivative is $\frac{\mathrm{d}y}{\mathrm{d}x}=18x^2$
b) $2$
c) $-6$
d) $f'(x)=2x$
e) $g'(a)=-16a^3$
f) $f'(1)=21$
g) $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{2\sqrt{x^3}}$
h) The gradient it $22$
i) The gradeint is $\frac34$