Referring to the AP Inter 1st Year Maths Study Material Chapter 12 Introduction to 3D Geometry Exercise 12d Solutions makes it easier to understand complex problems.
AP Inter 1st Year Maths Introduction to 3D Geometry Solutions Exercise 12d
Differentiate the functions with respect to x in Exercises 1 to 8.
Question 1.
sin(x2 + 5)
Solution:
Lety = sin (x2 + 5).
Differentiate w.r.t ‘x’
\(\frac{d y}{d x}\) = cos(x + 5) (2x)
⇒ \(\frac{d y}{d x}\) = 2x cos (x2 + 5)
Question 2.
cos(sin x)
Solution:
Let y = cos (sin x). Differentiate w.r.t ‘x’
\(\frac{d y}{d x}\) = sin(sin x) cosx
Question 3.
sin(ax + b)
Solution:
Lety = sin(ax + b).
Differentiate w.r.t ‘x’
\(\frac{d y}{d x}\) = cos(ax + b) (ax + b)
⇒ \(\frac{d y}{d x}\) = a cos (ax + b)
Question 4.
sec(tan (√x))
Solution:
Let y = sec (tan (√x)).
Differentiate w.r.t x’
\(\frac{d y}{d x}=\frac{d}{d x}\) sec(tan √x)
⇒ \(\frac{d y}{d x}\) = sec(tan √x) tan(tan √x) \(\frac{d}{d x}\)(tan √x)
= sec(tan √x) tan(tan √x) sec2 √x × \(\frac{1}{2 \sqrt{x}}\)
∴ \(\frac{d y}{d x}=\frac{\sec (\tan \sqrt{x}) \tan (\tan \sqrt{x}) \sec ^2 \sqrt{x}}{2 \sqrt{x}}\)
Question 5.
\(\frac{\sin (a x+b)}{\cos (c x+d)}\)
Solution:
Let y = \(\frac{\sin (a x+b)}{\cos (c x+d)}\)
Differentiate w.r.t ‘x’
\(\frac{d y}{d x}=\frac{\cos (c x+d) \frac{d}{d x} \sin (a x+b)-\sin (a x+b) \frac{d}{d x} \cos (c x+d)}{[\cos (c x+d)]^2}\)
Question 6.
cos x3. sin2(x5)
Solution:
Let y = cos x3 sin2 (x5).
Differentiate w.r.t ‘x’
\(\frac{1}{2}\) = cos x3 \(\frac{1}{2}\) sin2(x5) + sin2(x5) cos x3
since (sin2 x5) = 2 sin x5 sin x5 – 2 sin x5 cos x5 x5
= 10 x4 sin x5 cos x5
\(\frac{d}{d x}\) cos x3 = -sin x \(\frac{d}{d x}\)x3 = -3x2sin x3.
\(\frac{d y}{d x}\) = cos3 x(10x sin x5 cos x5) + sin2 x5(-3 x2 sin x3)
\(\frac{d y}{d x}\) = – 10x4 sin x5 cos x5 cos3 x – 3x2 sin x3 sin2 x5
Question 7.
cos √x
Solution:
Let y = cos √x.
Differentiate w.r.t ‘x’
\(\frac{d y}{d x}\) = -sin √x \(\frac{d}{d x}\)√x
⇒ \(\frac{d y}{d x}=\frac{-\sin \sqrt{x}}{2 \sqrt{x}}\)